Compound Poisson Approximation and Testing for Gene Clusters with Multigene Families

Remark 1

We can represent the measure μ_m as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{m} = \sum \nolimits_{i = 1}^{n}L_{i} \delta_{U_{(i)}}$$\end{document} , where δ_x denotes the Dirac measure in x and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{L}} = (L_{1}, \ldots, L_{n})$$\end{document} is a random vector independent of the U _(i) 's and uniformly distributed over the set of all possible labelings of the n orthologs: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \Lambda = \left\{{\bf \ell} = (\ell_{1}, \ldots, \ell_{n}) \in \left\{\frac {1} {\phi_{1}^{\prime}}, \ldots, \frac {1} {\phi_{J}^{\prime}} \right\}^{n} : \left | \left\{i: \ell_{i} = \frac {1} {\phi_{j}^{\prime}} \right\} \right | = g_{j} \phi_{j}^{\prime}, \forall j \right\}. \end{align*}\end{document}

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n_{\min}: \ = g_{1} = \mid \{i: \phi_{i} = \phi_{1}^{\prime} \} \mid$$\end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\phi_{1}^{\prime} = \min \{ \phi_{i}:i = 1, \ldots, m \}$$\end{document} . We let also \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$h_{\ast}: \ = \lceil h \phi_{1}^{\prime} \rceil$$\end{document} and we assume that n_min ≥ h_*, s.t. h_* is the minimal number of orthologs in a cluster of weight greater than h.

For every labeling \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda$$\end{document} and every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k = 1, \ldots, n$$\end{document} , let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} h_{k} ({\bf \ell}) : \ = \min \{d: \ell_{k} + \cdots + \ell_{k + d - 1} \geq h \} \end{align*}\end{document}

be the minimal number of orthologs in a cluster starting with the k-th ortholog so as to be of weight greater than h. Therefore, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} A_{k} \cap \{\textbf{\textit{L}} = {\bf \ell} \} = \{U_{(k + h_{k} ({\bf \ell}) - 1)} - U_{(k)} \leq r \} \cap \{\textbf{\textit{L}} = {\bf \ell} \}. \end{align*}\end{document}

Let h* := max _ℓ,k {h_k( ℓ )}.

We have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$h^{\ast} \leq \lceil h \phi_{J}^{\prime} \rceil$$\end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\phi_{J}^{\prime} = \max \{ \phi_{i}:i = 1, \ldots, m \}$$\end{document} .

Remark 2

We place ourselves in the asymptotic settings of m → ∞ , or equivalently, n → ∞ .

Note that we are in the case of a sum of indicators which are in a short-range dependence and a long-range (almost) independence. Because of the strong dependence between the neighboring indicators, the events A_k will tend to occur in clumps. Consequently, it seems reasonable to approach the distribution of W_m by a compound Poisson distribution CP(λ), with a “good” choice of the parameter λ.

We will quantify the error using the Kolmogorov distance. We recall that the Kolmogorov distance between two measures μ and ν on \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb R}_{+}$$\end{document} is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} d_{K} (\mu, \nu) = \sup_{k \in {\mathbb N}} \mid \mu ([k, \infty) - \nu ([k, \infty)) \mid. \end{align*}\end{document}

We will approximate our probability of interest \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb P} (W_{m} \geq1)$$\end{document} by the corresponding probability \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$CP ({\bf \lambda}) ([1, \infty)) = 1 - \exp \{ - \sum \limits_{i = 1}^{\infty} \lambda_{i} \}$$\end{document} for the compound Poisson distribution, with an error \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mid {\mathbb P} (W_{m} \geq1) - CP ({\bf \lambda}) ([1, \infty)) \mid \leq d_{K} ({\cal L} (W_{m}), CP ({\bf \lambda})). \end{align*}\end{document}

It is therefore sufficient to obtain bounds for the Kolmogorov distance between the two distributions. For bounding the Kolmogorov distance, we will use the Stein-Chen method for compound Poisson approximation.

The Stein-Chen method is a general method to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was originally formulated for normal approximations by Stein (1972), to obtain a bound for the Kolmogorov distance between the distribution of a sum of m-dependent sequence of random variables and a standard normal distribution. The Poisson approximation version was developed by Chen (1975). See also Stein (1986).

The Stein-Chen method for compound Poisson approximation was introduced by Barbour et al. (1992). In this article, we use, more precisely, the coupling approach developed by Roos (1993 a,b).

Theorem 1

Assume that for every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\alpha \in \Gamma$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$v \in{\cal V}_{\alpha}$$\end{document} we can construct, on the same probability space, the indicators \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{I_{\beta iv}^{\prime \prime} (\alpha), \beta \in \Gamma, i = 1, \ldots, \mid \Gamma_{\alpha}^{vs} \mid + 1 \}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{I_{\beta v}^{\prime} (\alpha), \beta \in \Gamma \}$$\end{document} in such a way that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\cal L} (I_{\beta iv}^{\prime \prime} (\alpha), \beta \in \Gamma) = {\cal L} (I_{\beta}, \beta \in \Gamma \mid {\textbf {\textit {I}}}_{\alpha}{\bf 1}_{\{Z_{\alpha} = i \}} = 1, V_{\alpha} = v), \forall i \tag {1} \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\cal L} (I_{\beta v}^{\prime} (\alpha), \beta \in \Gamma) & = {\cal L} (I_{\beta}, \beta \in \Gamma). \tag {2} \end{align*}\end{document}

Then, for all choices of the sets \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Gamma_{\alpha}^{vs}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Gamma_{\alpha}^{vw}$$\end{document} and for all bounded functions \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$g : g : {\mathbb N} \longrightarrow {\mathbb R}$$\end{document} , we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} d_{K} ({\cal L} (W), CP ({\hat{\bf \lambda}})) & \leq c_{{\cal K}} ({\hat{\bf \lambda}}) \left\{ \sum_{\alpha \in \Gamma} [({\mathbb E}I_{\alpha})^{2} + {\mathbb E}I_{\alpha}{\mathbb E} (U_{\alpha} + X_{\alpha}) + {\mathbb E} (I_{\alpha}X_{\alpha}) \right. \\ & \left. + \sum_{i = 1}^{\mid \Gamma_{\alpha}^{vs} \mid + 1} \sum_{\beta \in \Gamma_{\alpha}^{vw}}{\mathbb E} (I_{\alpha}{\bf 1}_{\{Z_{\alpha} = i \}} \theta_{\beta, \alpha, i} (V_{\alpha}))] + \sum_{i = \ell + 1}^{G + 1}i (i - 1) \lambda_{i} \right\}, \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$c_{\cal K} ({\hat{\bf \lambda}}) = \sup \nolimits_{A \in{\cal K}} \sup \nolimits_{i \geq1} \mid g_{{\hat{\bf \lambda}}, A} (i + 1) - g_{{\hat{\bf \lambda}}, A} (i) \mid$$\end{document} , with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal K} = \{ [k, \infty) : k \in {\mathbb N} \}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$g_{{\hat{\bf \lambda}}, A}$$\end{document} being the solution of the Stein-Chen equation for compound Poisson approximation (Roos, 1993a) and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\theta_{\beta, \alpha, i} (v) = {\mathbb E} \mid I_{\beta iv}^{\prime \prime} (\alpha) - I_{\beta v}^{\prime} (\alpha) \mid$$\end{document} .

Barbour et al. (2000) showed that, if the following condition is fulfilled: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \lambda_{1} \geq 2 \lambda_{2} \geq 3 \lambda_{3} \geq \cdots, \tag{3} \end{align*}\end{document}

then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} c_{\cal K} ({\bf \lambda}) \leq \min \left\{\frac {1} {2}, \frac {1} {\lambda_{1} + 1} \right\}. \tag {4} \end{align*}\end{document}

In the next section, we apply this method to our W_m, using Theorem 1.

Notation 1

For a sequence (a_j)_j≥1 we will denote \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$a_{i, k} \ : = a_{i} + \cdots + a_{i + k - 1}$$\end{document} .

For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k \in \{1, \ldots, n \}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda$$\end{document} , we have A_k ∩ { L  =  ℓ } = {S_{k,hk( ℓ )−1} ≤ r}.

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k \in \{1, \ldots, n \}$$\end{document} be fixed. The indicators appearing in the expression of Z_k are those from I_k−h*+2 to I_k+h*−2. Consequently, if L  =  ℓ , the spacings appearing in the expression of Z_k are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{k - h_{\ast} + 2}, \ldots$$\end{document} , S_{k+h*+hk+h*−2( ℓ )−4}.

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$V_{k} : \ = (\textbf{\textit{L}}, S_{k - h_{\ast} + 2}, \ldots, S_{k + h_{\ast} + h^{\ast} - 4})$$\end{document} . Note that V_k contains all the spacings which may appear in Z_k, for different values of ℓ .

For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$v = ({\bf \ell}, z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5})$$\end{document} , with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda, z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5} > 0$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$z_{1} + \cdots + z_{2h_{\ast} + h^{\ast} - 5} < 1$$\end{document} , we will construct on the same probability space the indicators \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{I_{jiv}^{\prime \prime} (k), j = 1, \ldots, n \}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{I_{j}^{\prime} (k), j = 1, \ldots, n \}$$\end{document} (not depending on v) verifying the relations (1) and (2) in Theorem 1.

Note that the event {I_k1_{Zk=i} = 1} is V_k - measurable and thus, for the condition (1) to be fulfilled, it suffices to construct the family of indicators \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{I_{j}^{\prime} (k), j = 1, \ldots, n \}$$\end{document} (not depending on i), s.t. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\cal L} (I_{jv}^{\prime \prime} (k), j = 1, \ldots, n) = {\cal L} (I_{j}, j = 1, \ldots, n \mid V_{k} = ({\bf \ell}, z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5})). \end{align*}\end{document}

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$U_{1}^{\prime}, \ldots, U_{n}^{\prime}$$\end{document} be r.v.'s independent on L and such that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal L} (U_{1}^{\prime}, \ldots, U_{n}^{\prime}) = {\cal L} (U_{(1)}, \ldots, U_{(n)})$$\end{document} .

Define the corresponding spacings \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{j}^{\prime} = U_{j + 1}^{\prime} - U_{j}^{\prime}, \forall j = 1, \ldots, n$$\end{document} (with the circular convention \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$U_{n + 1}^{\prime} = U_1^{\prime}$$\end{document} ). We thus have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal L} (S_{1}^{\prime}, \ldots, S_{n}^{\prime}) = {\cal L} (S_{1}, \ldots, S_{n})$$\end{document} .

For \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$v = ({\bf \ell}, z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5})$$\end{document} with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda, z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5} > 0$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$z_{1} + \cdots + z_{2h_{\ast} + h^{\ast} - 5} < 1$$\end{document} , we let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} & S_{j}^{\prime \prime} = \frac {1 - \sum \limits_{i = 1}^{2h_{\ast} + h^{\ast} - 5} z_{i}} {1 - \sum \limits_{i = k - h_{\ast} + 2}^{k + h_{\ast} + h^{\ast} - 4} S_{i}^{\prime}} S_{j}^{\prime}, j \in \{ 1, \ldots, n \} \backslash \{ k - h_{\ast} + 2, \ldots, k + h_{\ast} + h^{\ast} - 4 \}, \\ & S_{k - h_{\ast} + 2}^{\prime \prime} = z_{1}, \ldots, S_{k + h_{\ast} + h^{\ast} - 4}^{\prime \prime} = z_{2h_{\ast} + h^{\ast} - 5}. \end{align*}\end{document}

Note that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\cal L} (S_{1}^{\prime \prime}, \ldots, S_{n}^{\prime \prime}) = {\cal L} (S_{1}, \ldots, S_{n} \mid S_{k - h_{\ast} + 2} = z_{1}, \ldots, S_{k + h_{\ast} + h^{\ast} - 4} = z_{2h_{\ast} + h^{\ast} - 5}). \end{align*}\end{document}

Let also \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{m}^{\prime}: \ = \sum \nolimits_{i = 1}^{n}L_{i} \delta_{U_{i}^{\prime}}$$\end{document} .

For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$j \in \{1, \ldots, n \}$$\end{document} we construct the indicators needed in Theorem 1 as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} I_{j}^{\prime} (k) : \ = {\bf 1}_{\{ \mu_{m}^{\prime} ([U_{j}^{\prime}, U_{j}^{\prime} + r]) \geq h \}}, I_{jv}^{\prime \prime} (k) : = {\bf 1}_{\{S_{j}^{\prime \prime} + \cdots + S_{j + h_{j} ({\bf \ell}) - 2}^{\prime \prime} \leq r \}}. \end{align*}\end{document}

It is easy to see that the indicators defined above verify the conditions (1) and (2). It remains to compute all the quantities appearing in the error bound in Theorem 1.

The canonical choice for the parameters of the compound Poisson distribution is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\boldsymbol \lambda} = \sum \nolimits_{i = 1}^{2h_{\ast} - 3} \lambda_{i} \delta_{i},$$\end{document} , with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\lambda_{i} = \frac {1} {i} \sum \nolimits_{k = 1}^{n} {\mathbb E} (I_{k} {\bf 1}_\{ Z_{k} = i \})$$\end{document} .

In our approximation we will use only half of the parameters, by truncating at ℓ = h_* − 1. Instead of λ we will use \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\hat{\bf \lambda}} = \sum \nolimits_{i = 1}^{h_{\ast} - 1} \hat{\lambda}_{i} \delta_{i}$$\end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat{\lambda}_{i} = \lambda_{i}$$\end{document} for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$i = 2, \ldots, h_{\ast} - 1$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat{\lambda}_{1} = {\mathbb E} (W_{m}) - \sum \nolimits_{i = 2}^{h_{\ast} - 1} \lambda_{i} = \lambda_{1} + \sum \nolimits_{i = h_{\ast}}^{2h_{\ast} - 3}i \lambda_{i}$$\end{document} .

We will approximate the probability of interest \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb P} (W_{m} \geq1)$$\end{document} by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} p : \ = 1 - \exp \left\{ - \sum_{i = 1}^{h_{\ast} - 1} \hat{\lambda}_{i} \right\}. \end{align*}\end{document}

Remark 3

As the indicators \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{I_{jv}^{\prime \prime} (k), j = 1, \ldots, n \}$$\end{document} do not depend on i, also the term \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\theta_{j, k} (v) = {\mathbb E} \left|I_{jv}^{\prime \prime} (\alpha) - I_{jv}^{\prime} (\alpha) \right|$$\end{document} appearing in Theorem 1 does not depend on i and thus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} d_{K} ({\cal L} (W_{m}), CP ({\hat{\bf \lambda}})) \leq c_{K} ({\hat{\bf \lambda}}) \left\{ \sum_{k = 1}^{n} \{ ({\mathbb E}I_{k})^{2} + {\mathbb E}I_{k}{\mathbb E} (U_{k} + X_{k}) + {\mathbb E} (I_{k}X_{k}) \right. \\ \left. + \sum_{j \in \Gamma_{k}^{vw}}{\mathbb E} (I_{k} \theta_{j, k} (V_{k})) \} + \sum_{i = h_{\ast}}^{2h_{\ast} - 3}i (i - 1) \lambda_{i} \right\} , \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$U_{k} = \sum \nolimits_{j = k - h_{\ast} + 2}^{k + h_{\ast} - 2}I_{j} - I_{k}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$X_{k} = \sum \nolimits_{j = k - 2h^{\ast} + 4}^{k - h_{\ast} + 1}I_{j} + \sum \nolimits_{j = k + h_{\ast} - 1}^{k + 2h^{\ast} - 1}I_{j}$$\end{document} .

Using classic results on uniform spacings (Pyke, 1965, 1972), one can easily prove

Lemma 2

For fixed k, assume that n → ∞ , r → 0 s.t. nr → 0. Then, uniformly with respect to 0 < nr < 1, we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (S_{1, k} \leq r) = \frac {(nr)^{k}} {k!} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right) \end{align*}\end{document}

and for fixed i and j, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} if \ i < k: {\mathbb P} (S_{1, i} \leq r, S_{k, j} \leq r) & = \frac {(nr)^{i + j}} {i!j!} (1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr)), \\ if \ i \geq k: {\mathbb P} (S_{1, i} \leq r, S_{k, j} \leq r) & = \frac {(2k - i + j - 2) !} {(k + j - 1) ! (k - 1) ! (k - i + j - 1) !} (nr)^{k + j - 1} \\ & \quad \ \times \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right). \end{align*}\end{document}

For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k = 1, \ldots, n$$\end{document} , we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{k}) = \frac {1} {\mid \Lambda \mid} \sum \nolimits_{{\bf \ell} \in \Lambda} {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r)$$\end{document} .

For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda$$\end{document} , from Lemma 1 and the exchangeability of the spacings, we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r) = {\mathbb P} (S_{1, h_{k} ({\bf \ell}) - 1} \leq r) = \frac {(nr)^{h_{k} ({\bf \ell}) - 1}} {(h_{k} ({\bf \ell}) - 1) !} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right). \end{align*}\end{document}

We hence obtain, for 0 < nr < 1, the following upper bound: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb E} (I_{k}) \leq \frac {(nr)^{h_{\ast} - 1}} {(h_{\ast} - 1) !} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right). \tag {5} \end{align*}\end{document}

We make the following (biologically realistic) assumption on the data.

Assumption 1

We assume that we have n_min ≈ n, i.e., \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n_{\min} = \alpha n \left(1 + {\cal O} \left(\frac {1} {n} \right) \right)$$\end{document} , with α ≤ 1 fixed.

Based on Assumption 1, we obtain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mid \{{\bf \ell} \in \Lambda:h_{1} ({\bf \ell}) = h_{\ast} \} \mid \asymp \mid \Lambda \mid. \tag{6} \end{align*}\end{document}

This further implies that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{k}) \asymp (nr)^{h_{\ast} - 1}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (W_{m}) \asymp n (nr)^{h_{\ast} - 1}$$\end{document} .

Let k < j. We have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{k} I_{j}) = \frac {1} {\mid \Lambda \mid} \sum \nolimits_{{\bf \ell} \in \Lambda} {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r, S_{j, h_{j} ({\bf \ell}) - 1} \leq r)$$\end{document} , where for each \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda$$\end{document} , using Lemma 1, we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} & {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r, S_{j, h_{j} ({\bf \ell}) - 1} \leq r) = {\mathbb P} (S_{1, h_{k} ({\bf \ell}) - 1} \leq r, S_{j - k + 1, h_{j} ({\bf \ell}) - 1} \leq r) \\ & = \frac {(2 (j - k) + h_{j} ({\bf \ell}) - h_{k} ({\bf \ell})) !} {(j - k) ! (j - k + h_{j} ({\bf \ell}) - h_{k} ({\bf \ell})) ! (j - k + h_{j} ({\bf \ell}) - 1) !} (nr)^{j - k + h_{j} ({\bf \ell}) - 1} \\ & \quad \times \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right), \tag {7} \\ &\quad \ {\rm if} \ k < j \leq k + h_{k} ({\bf \ell}) - 2 \ (\hbox {\rm we say that} \ the \ two \ clusters \ intersect) \\ & = \frac {1} {(h_{k} ({\bf \ell}) - 1) ! (h_{j} ({\bf \ell}) - 1) !} (nr)^{h_{k} ({\bf \ell}) + h_{j} ({\bf \ell}) - 2} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right), \tag {8} \\ & \quad \ \ {\rm if} \ j > k + h_{k} ({\bf \ell}) - 2 \ (\hbox {\rm we say that} \ the \ two \ clusters \ do \ not \ intersect). \end{align*}\end{document}

From Assumption 1, we can also obtain that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \mid \{{\bf \ell} \in \Lambda:h_{k} ({\bf \ell}) = h_{\ast}, h_{j} ({\bf \ell}) = h_{\ast} \} \mid \asymp \mid \Lambda \mid. \tag{9} \end{align*}\end{document}

Next we will estimate the error terms appearing in Theorem 1. We have

Proposition 3

Assume that n → ∞ , r → 0 s.t. nr → 0 and n_min ≈ n. Then, uniformly in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\frac {1} {n} \leq nr < 1$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n > 2 (2h_{\ast} + h^{\ast} - 4) \vee \exp \left\{\frac {4 (h_{\ast} + h^{\ast} - 5)} {3 (h_{\ast} - 1) + h^{\ast}} \right\}$$\end{document} , we have the following estimates: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} & (a) \ \sum_{k = 1}^{n} ({\mathbb E} I_{k})^{2} \leq \frac {n (nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right) ; \\ & (b) \ \sum_{k = 1}^{n} {\mathbb E} (I_{k}) {\mathbb E} (U_{k} + X_{k}) \leq 4 (h^{\ast} - 2) \frac {n (nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} (1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr)) ; \\ & (c) \ \sum_{k = 1}^{n} {\mathbb E} (I_{k} X_{k}) \leq 2 (2h^{\ast} - h_{\ast} - 2) \frac {n (nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right) ; \\ & (d) \ \sum_{k = 1}^{n} \sum_{j \in \Gamma_{k}^{vw}} {\mathbb E} (I_{k} \theta_{j, k} (V_{k})) \leq 2 (h_{\ast} - 1) \{ 2h_{\ast} + h^{\ast} - 5 + 2^{h_{\ast} - 2} (h_{\ast} + h^{\ast} - 4) \} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \times \frac {n (nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right) ; \\ & (e) \ \sum \limits_{i = h_{\ast}}^{2h_{\ast} - 3} i (i - 1) \lambda_{i} \leq (h_{\ast} - 2) 2^{2h_{\ast} - 5} \frac {n (nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right). \end{align*}\end{document}

Proof. The bounds in (a) and (b) follow easily using (5).  ▪

Proof of (c). We have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \sum_{k = 1}^{n}{\mathbb E} (I_{k}X_{k}) = \sum_{k = 1}^{n} \left\{ \sum_{j = k - 2h^{\ast} + 4}^{k - h_{\ast} + 1}{\mathbb E} (I_{j}I_{k}) + \sum_{j = k + h_{\ast} - 1}^{k + 2h^{\ast} - 4}{\mathbb E} (I_{k}I_{j}) \right\}, \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{k} I_{j}) = \frac {1} {\mid \Lambda \mid} \sum \nolimits_{{\bf \ell} \in \Lambda} {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r, S_{j, h_{j} ({\bf \ell}) - 1} \leq r)$$\end{document} .

For j = k + h_* − 1, using Lemma 1,

– if h_j( ℓ ) = h_* and h_k( ℓ ) = h_*, then the two clusters do not intersect and, from (8), we obtain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r, S_{j, h_{j} ({\bf \ell}) - 1} \leq r) = \frac {(nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right) ; \end{align*}\end{document}

– if h_j( ℓ ) = h_* and h_k( ℓ ) > h_*, then the two clusters intersect and, using (7), we obtain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r, S_{j, h_{j} ({\bf \ell}) - 1} \leq r) \leq \frac {1} {2} \frac {(nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right) ; \end{align*}\end{document}

– for every other ℓ we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb P} (S_{k, h_{k} ({\bf \ell}) - 1} \leq r, S_{j, h_{j} ({\bf \ell}) - 1} \leq r) = (nr)^{2 (h_{\ast} - 1)}{\cal O} (nr)$$\end{document} .

It follows from (9) that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{k} I_{k + h_{\ast} - 1}) \leq \frac {(nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right)$$\end{document} .

The other cases for j can be treated in a similar manner and the upper bound stated in (c) easily follows.  ▪

Proof of (d). We will condition on the r.v. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$V_{k} = (\textbf{\textit{L}}, S_{k - h_{\ast} + 2}, \ldots, S_{k + h_{\ast} + h^{\ast} - 4})$$\end{document} . Given that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$V_{k} = ({\bf \ell}, z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5})$$\end{document} , we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$I_{k} = {\bf 1}_{\{z_{h_{\ast} - 1, h_{k} ({\bf \ell}) - 1} \leq r \}}$$\end{document} (hence deterministic) and thus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \sum_{k = 1}^{n} \sum_{j \in \Gamma_{k}^{vw}} {\mathbb E} (I_{k} \theta_{j, k} (V_{k})) = \sum_{k = 1}^{n} \sum_{j \in \Gamma_{k}^{vw}} \frac {1} {\mid \Lambda \mid} \sum_{{\bf \ell} \in \Lambda} d (k, j, {\bf \ell}), \end{align*}\end{document}

where for each \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k = 1, \ldots, n$$\end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$j \in \Gamma_{k}^{vw}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\bf \ell} \in \Lambda$$\end{document} , we let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} d (k, j, {\bf \ell}) : \ & = {\mathbb E} [I_{k} \theta_{j, k} (V_{k}) \mid \textbf{\textit{L}} = {\bf \ell}] = d_{1} (k, j, {\bf \ell}) + d_{2} (k, j, {\bf \ell}), \\ d_{1} (k, j, {\bf \ell}) : \ & = \int {\bf 1}_{\{z_{h_{\ast} - 1, h_{k} ({\bf \ell}) - 1} \leq r \}}{\mathbb P} (S_{j, h_{j} ({\bf \ell}) - 1}^{\prime \prime} \leq r, S_{j, h_{j} ({\bf \ell}) - 1}^{\prime} > r) \\ & \quad \quad dF (z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5}), \\ d_{2} (k, j, {\bf \ell}) : \ & = \int{\bf 1}_{\{z_{h_{\ast} - 1, h_{k} ({\bf \ell}) - 1} \leq r \}}{\mathbb P} (S_{j, h_{j} ({\bf \ell}) - 1}^{\prime \prime} > r, S_{j, h_{j} ({\bf \ell}) - 1}^{\prime} \leq r) \\ & \quad \quad dF (z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5}), \end{align*}\end{document}

with F being the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$(S_{k - h_{\ast} + 2}, \ldots, S_{k + h_{\ast} + h^{\ast} - 4})$$\end{document} .

We further decompose \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$d_{1} (k, j, {\bf \ell}) : = d_{1}^{\prime} (k, j, {\bf \ell}) + d_{1}^{\prime \prime} (k, j, {\bf \ell})$$\end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} d_{1}^{\prime} (k, j, {\bf \ell}) : \ & = \int {\bf 1}_{\{z_{h_{\ast} - 1, h_{k} ({\bf \ell}) - 1} \leq r \}}{\bf 1}_{\{z_{1, 2h_{\ast} + h^{\ast} - 5} > ar \}} \\ & \qquad \ \times {\mathbb P} (S_{j, h_{j} ({\bf \ell}) - 1}^{\prime \prime} \leq r, S_{j, h_{j} ({\bf \ell}) - 1}^{\prime} > r) dF (z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5}), \\ d_{1}^{\prime \prime} (k, j, {\bf \ell}) : \ & = \int {\bf 1}_{\{z_{h_{\ast} - 1, h_{k} ({\bf \ell}) - 1} \leq r \}} {\bf 1}_{\{z_{1, 2h_{\ast} + h^{\ast} - 5} \leq ar \}} \\ & \qquad \ \times {\mathbb P} (S_{j, h_{j} ({\bf \ell}) - 1}^{\prime \prime} \leq r, S_{j, h_{j} ({\bf \ell}) - 1}^{\prime} > r) dF (z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5}), \end{align*}\end{document}

with a = a(n) to be chosen a little further.  ▪

We will simplify the notation by writing h_k instead of h_k( ℓ ). We have: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} d_{1}^{\prime} (k, j, {\bf \ell}) & \leq \int {\bf 1}_\{ {z_{h_{\ast} - 1, h_{k} - 1} < r} \} {\bf 1}_\{ {z_{1, 2h_{\ast} + h^{\ast} - 5} > ar} \} dF (z_{1}, \ldots, z_{2h_{\ast} + h^{\ast} - 5}) \\ & = \int_{0}^{r} \frac {n (nu)^{h_{k} - 2}} {(h_{k} - 2) !} \int_{ar - u}^{1} \frac {n (nv)^{2h_{\ast} + h^{\ast} - h_{k} - 5}} {(2h_{\ast} + h^{\ast} - h_{k} - 5) !} (1 - u - v)^{n - (2h_{\ast} + h^{\ast} - 4)} \\ & \quad \times \left(1 + {\cal O} \left(\frac {1} {n} \right) \right) dvdu \\ & \leq \int_{0}^{nr} \frac {x^{h_{k} - 2}} {(h_{k} - 2) !} e^{- x / 2} \int_{anr - x}^{n} \frac {y^{2h_{\ast} + h^{\ast} - h_{k} - 5}} {(2h_{\ast} + h^{\ast} - h_{k} - 5) !} e^{- y / 2} \\ & \quad \times \left(1 + {\cal O} \left(\frac {1} {n} \right) \right) dydx \ (\hbox {\rm by a change of variable} + {\rm Lemma} \ 3) \\ & \leq 2^{2h_{\ast} + h^{\ast} - 5} \int_{0}^{nr / 2} \frac {z^{h_{k} - 2}} {(h_{k} - 2) !} e^{- z} \int_{anr / 2 - z}^{\infty} \frac {t^{2h_{\ast} + h^{\ast} - h_{k} - 5}} {(2h_{\ast} + h^{\ast} - h_{k} - 5) !} e^{- t} \\ & \quad \times \left(1 + {\cal O} \left(\frac {1} {n} \right) \right) dydx \ \\ & \leq \frac {4 (nr)^{h_{k} - 1}} {(h_{k} - 1) !} \frac {(anr)^{2h_{\ast} + h^{\ast} - h_{k} - 5}} {(2h_{\ast} + h^{\ast} - h_{k} - 5) !} e^{- anr / 2} \left(1 + {\cal O} \left(\frac {1} {n} \right) \right) (\hbox {\rm by Lemma} \ 2) \\ & \leq \frac {4 (nr)^{h_{k} + h_{\ast} - 2}} {(h_{\ast} - 1) !} \frac {1} {n} \left[\frac {n} {(nr)^{h_{\ast} - 1}} \frac {(anr)^{2h_{\ast} + h^{\ast} - h_{k} - 5}} {(2h_{\ast} + h^{\ast} - h_{k} - 5) !} e^{- anr / 2} \right] \\ & \quad \times \left(1 + {\cal O} \left(\frac {1} {n} \right) \right) \\ & \leq \frac {1} {n} (nr)^{2 (h_{\ast} - 1)} {\cal O} (nr), \end{align*}\end{document}

if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\frac {1} {n} \leq nr$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$(3h_{\ast} + h^{\ast} - 3) \log n \leq anr \leq \sqrt{n}$$\end{document} , entailing that

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n (anr)^{2h_{\ast} + h^{\ast} - h_{k} - 5}e^{- anr / 2} \leq (nr)^{h_{\ast}}$$\end{document} , and if moreover a > 1, nr < 1 and

anr > 4(2h_* + h* − 4) for applying Lemma 2 and Lemma 3. The last inequality is hence valid for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} 4 (2h_{\ast} + h^{\ast} - 4) \vee (3h_{\ast} + h^{\ast} - 3) \log n \leq anr \leq \sqrt {n} \ {\rm and} \ \frac {1} {n} \leq nr < 1 \end{align*}\end{document}

In a similar manner we can bound \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$d_{1}^{\prime \prime} (k, j, {\bf \ell})$$\end{document} , then decompose and bound d₂(k, j, ℓ ).

We finally obtain that, if we take \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$a: \ = \frac {(3h_{\ast} + h^{\ast} - 3) \log n} {nr}$$\end{document} then, uniformly in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\frac {1} {n} \leq nr < 1$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n > 4 (2h_{\ast} + h^{\ast} - 4) \vee \exp \left\{\frac {4 (2h_{\ast} + h^{\ast} - 4)} {3h_{\ast} + h^{\ast} - 3} \right\}$$\end{document} , we have the upper bound stated in (d).

Proof of (e). For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k = 1, \ldots, n$$\end{document} we let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\cal C}_{ik}$$\end{document} denote the class of all the subsets of size i − 1 of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Gamma_{k}^{vs} = \{k - h_{\ast} + 2, \ldots, k - 1, k + 1, \ldots, k + h_{\ast} - 2 \}$$\end{document} . We obtain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} i \lambda_{i} = \sum_{k = 1}^{n} \sum_{C \in{\cal C}_{ik}}{\mathbb E} \left(I_{k} \prod \limits_{t \in C}I_{t} \prod \limits_{t \in \Gamma_{k}^{vs} \setminus C} (1 - I_{t}) \right) \leq \sum_{k = 1}^{n} \sum_{C \in{\cal C}_{ik}}{\mathbb E} (I_{\inf C}I_{\sup C}). \end{align*}\end{document}

For every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$k = 1, \ldots, n$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$C \in {\cal C}_{ik}$$\end{document} we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb E} (I_{\inf C} I_{\sup C}) = \frac {1} {\mid \Lambda \mid} \sum_{{\bf \ell} \in \Lambda} {\mathbb P} (S_{\inf C, h_{\inf C} ({\bf \ell}) - 1} \leq r, S_{\sup C, h_{\sup C} ({\bf \ell}) - 1} \leq r) \end{align*}\end{document}

and h_* − 1 ≤ i − 1 ≤ sup C − inf C.

If h_{inf C}( ℓ ) = h_{sup C}( ℓ ) = h_*, then the two clusters do not intersect and we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (S_{\inf C, h_{\inf C} ({\bf \ell}) - 1} \leq r, S_{\sup C, h_{\sup C} ({\bf \ell}) - 1} \leq r) = \frac {(nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right). \end{align*}\end{document}

It follows that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{\inf C} I_{\sup C}) \leq \frac {(nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right)$$\end{document} and hence \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} i \lambda_{i} \leq \left(\begin {matrix} 2 (h_{\ast} - 2) \\ i - 1\end {matrix} \right) \frac {n (nr)^{2 (h_{\ast} - 1)}} {[(h_{\ast} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right). \end{align*}\end{document}

The bound stated in (e) then follows from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \sum_{i = h_{\ast}}^{2h_{\ast} - 3} (i - 1) \left(\begin{matrix}2 (h_{\ast} - 2) \\ i - 1\end{matrix} \right) = (2h_{\ast} - 4) \sum_{j = h_{\ast} - 2}^{2h_{\ast} - 5} \left(\begin{matrix}2h_{\ast} - 5 \\ j\end{matrix} \right) = (h_{\ast} - 2) 2^{2h_{\ast} - 5}. \end{align*}\end{document}   ▪

We have used the following two elementary lemmas that we state without proof. For a proof of Lemma 4, see Grusea (2008), and for a proof of Lemma 5, see Roos (1993a).

Lemma 4

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$X_{1}, \ldots, X_{n}$$\end{document} be i.i.d. r.v.'s with distribution Exp(1) and let i, k ≥ 1 s.t. i + k − 1 ≤ n. Then, uniformly in α ≥ 1, β < 1, αβ > 2(n − k − 1), we have the following inequality: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (X_{1, n} > \alpha \beta, X_{i, k} < \beta) \leq2 \frac {\beta^{k}} {k!} \frac {(\alpha \beta)^{n - k - 1}} {(n - k - 1) !} e^{- \alpha \beta}. \end{align*}\end{document}

Lemma 5

For 0 ≤ x ≤ 1 and n ≥ 2(m + 1) we have (1 − x)^n−(m+1) ≤ e^−nx/2.

In the following lemma, we show that the chosen parameters for the approximating compound Poisson distribution verify the relation (3), and hence we can use the bound (4) of Barbour et al. (2000).

Lemma 6

If 0 < nr < 1 and n_min ≈ n, then for every \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$i \in \{1, \ldots, h_* - 1 \}$$\end{document} we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat{\lambda}_{i} \asymp n (nr)^{i + h_{*} - 2}$$\end{document} . If n_min ≈ n and nr ≤ γ, where γ is a fixed constant γ < 1, then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$i \hat{\lambda}_{\,i} \geq (i + 1) \hat{\lambda}_{i + 1}, \forall i$$\end{document} .

Proof. We have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$i \lambda_{i} = \sum \nolimits_{k = 1}^{n}{\mathbb E} (I_{k}{\bf 1}_{\{Z_{k} = i \}})$$\end{document} . One can easily show that the leading terms in the expression of iλ_i are those which are expectations of products of i consecutive indicators. For a term with i consecutive indicators, of the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb E} (I_{j} \cdots I_{j + i - 1}) = \frac {1} {\mid \Lambda \mid} \sum_{{\bf \ell} \in \Lambda} {\mathbb E} (I_{j} \cdots I_{j + i - 1} \mid \textbf {\textit {L}} = {\bf \ell}), \end{align*}\end{document}

we have that for each ℓ the extreme clusters intersect (because of the fact that j + i − 1 ≤ j + h_j( ℓ ) − 2, as i < h_* ≤ h_j( ℓ ), ∀j) and hence \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} {\mathbb P} (S_{j} + \cdots + S_{j + i - 1 + h_{j + i - 1} ({\bf \ell}) - 2} \leq r) & \leq{\mathbb E} (I_{j} \cdots I_{j + i - 1} \mid {\bf L} = {\bf \ell}) \\ & \leq{\mathbb E} (I_{j}I_{j + i - 1} \mid {\bf L} = {\bf \ell}), \end{align*}\end{document}

implying that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{j} \cdots I_{j + i - 1} \mid \textbf{\textit{L}} = {\bf \ell} \asymp (nr)^{i + h_{j + i - 1} ({\bf \ell}) - 2}.$$\end{document} .

Using (6) we obtain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (I_{j} \cdots I_{j + i - 1}) \asymp (nr)^{i + h_{*} - 2}, \forall j$$\end{document} .

The results in the statement easily follow.  ▪

For the detailed proofs of Proposition 1 and Lemma 2, see Grusea (2008).

From Proposition 1 and Lemma 4, together with Theorem 1 and relation (4), we obtain the following upper bound on the error of approximating \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb P} (W_{m} \geq1)$$\end{document} by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$p = 1 - \exp \{ - \sum \nolimits_{i = 1}^{h_* - 1} \hat{\lambda}_{i} \}$$\end{document} .

Theorem 7

Suppose that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n \longrightarrow \infty$$\end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$r \longrightarrow0$$\end{document} and n_min ≈ n. Then, uniformly in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\frac {1} {n} \leq nr < 1$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n > 2 (2h_{*} + h^{*} - 4) \vee \exp \left\{\frac {4 (2h_{*} + h^{*} - 4)} {3 (h_{*} - 1) + h^{*}} \right\}$$\end{document} , we have: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \left| {\mathbb P} (W_{m} \geq1) - p \right| \leq C \frac {n (nr)^{2 (h_{*} - 1)}} {[(h_{*} - 1) !]^{2}} \left(1 + {\cal O} \left(\frac {1} {n} \right) + {\cal O} (nr) \right), \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$C = 4h^{*} - h_{*} - 6 + (h_{*} - 1) \{2h_{*} + h^{*} - 5 + 2^{h_{*} - 2} (h_{*} + h^{*} - 4) \} + (h_{*} - 2) 2^{2h_{*} - 6}$$\end{document} .

Moreover, if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\mathbb E} (W_{m}) = \pi_{\infty}$$\end{document} is held constant when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n \longrightarrow \infty$$\end{document} , then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \left| {\mathbb P} (W_{m} \geq1) - p \right| = {\cal O} \left(\frac {1} {n} \right). \end{align*}\end{document}