Prediction of a structural instability in stochastic process

The idea of predicting the case, when the considered long-term ARMA model, fitted to the observed time series tends to become unstable because of deep changes in the structural stability of data, is developed in this paper. The aim is to predict a possible unstable regime of the process {X_t,t∈T}τ-steps in advance before it will express itself by a high level crossing or large variance of an output variable X_t. The problem is solved here for locally stationary AR(p) sequences {X_t,t∈T}, whose estimated parameters can reach critical sets located at the boundary of the stability area. An alarm function and an alarm set are fitted here to predict catastrophic failures in systems output τ units in advance for given τ>0 and a confidence level γ. The probability of false alarm is derived explicitly for AR(1) depending on τ,γ and N – the number of the last observations of {X_t}.