On lacunary generalized statistical convergent complex uncertain triple sequence

1 Introduction

We see that classical measure, satisfactory nonnegativity and countable additivity, is commonly used both in every-day life and mathematical theory; nonetheless, generally the measure used in real life events has no countable additivity. Researchers suggested different measures for real life events, such as capacities and fuzzy measures. However, both capacity and fuzzy measures point out continuity rather than self-duality and countable subadditivity. Since self-duality and countable subadditivity are significant both in theory and practice, Liu [15] introduced a self-duality measure, uncertain measure, which is a set function fulfilling normality, monotonicity, self-duality, and countable subadditivity axioms. In actual life, various types of uncertainty exist, such as randomness, fuzziness, and uncertainty which consists of both randomness and fuzziness. Probability measure is used to define a random event. To measure fuzzy events, Zadeh [34] initiated possibility measure but possibility measure does not have self duality. Thus, Liu and Liu [17] introduced a self-dual measure, the credibility measure. An undeniable foundation for credibility theory was given in Liu [15]. From the beginning, the uncertainty theory has been consistently studied, developed and has widespread application (see [16, 20]). Clearly, classical measure, probability measure and credibility measure are special cases of uncertain measures. But possibility measure is not an uncertain measure. Hence the elements of uncertain measure can also be applied to classical measure, probability measure and credibility measure.

Complex uncertain variables are measurable functions from uncertainty spaces to the set of complex numbers. Convergence of sequences always plays a crucial role in different theory of mathematics. The convergence of complex uncertain sequence was first introduced by Chen et al. [1]. Studies on convergence of sequences of uncertain variables are due to You [33]. The concept of statistical convergence, which is an extension of usual idea of convergence, was introduced by Fast [7] for real and complex number sequences. Statistical convergence has several applications in different fields of mathematics like number theory, trigonometric series, summability theory, statistics and probability theory, measure theory, optimization, approximation theory, rough set theory, hopfield neural networks and fuzzy sets. The concept of convergence of sequences of numbers has been extended by several researcher ([28–30]). The study of statistical convergence in triple sequence has been initiated by Şahiner et al. [23]. Fridy and Orhan [10] defined the concept of lacunary statistical convergence. Kostyrko et al. [13] extended the notion of statistical convergence to ideal convergence, and established some basic theorems. On the other hand, the new form of convergence called I -statistical convergence has been introduced in [2]. Recently lots of interesting developments have occurred in I -statistical convergence and related topics (see [21, 32])

Since sequence convergence plays a very important role in the fundamental theory of mathematics, there are many convergence concepts in classical measure theory, probability theory and credibility theory, and the relationships between them are discussed. The interested reader may consult Liu [14, 15], Zhu and Liu [35], Tripathy and Nath [26] and Das et al. [3]. Inspired by this, in this paper, a further investigation into the mathematical properties of uncertain triple sequences will be made. Section 2 recalls some definitions and theorems in uncertainty and summability theory. In Section 3, we analyze some lacunary I -statistical convergence concepts of complex uncertain triple sequences such as lacunary I -statistical convergence in measure, lacunary I -statistical convergence in mean, lacunary I -statistical convergence in distribution and with respect to almost surely and examine the relationships among them by several theorems and examples. Finally, some conclusions are given in Section 4.

2 Preliminaries

We recall the following basic concepts from [9, 15]. They will be needed in the course of the paper.

Definition 1. ([15]) On a non-empty set Γ, L is a σ-algebra. M is a set function which is called to be an uncertain measure when it ensures the undermentioned axioms:

(1 ^o ) M { Γ } = 1 (normality axiom);

(2 ^o ) M { Λ } + M { Λ c } = 1 for any Λ ∈ L (duality axiom);

(3 ^o ) We get M { ⋃ j = 1 ∞ λ j } ≤ ∑ j = 1 ∞ M { λ j }(2.1) for all countable sequence of { λ j } ∈ L (subadditivity axiom).

The triplet ( Γ , L , M ) is called an uncertainty space. Each element Λ is named an event in L . Liu [15] defined a product uncertain measure to get an uncertain measure of compound event as follows:

(4 ^o ) For s = 1, 2, 3, . . . , let ( Γ s , L s , M s ) be uncertainty space. The product uncertain measure M is a measure such that M { ∏ s = 1 ∞ Λ s } = ⋀ s = 1 ∞ M s { Λ s } , for s = 1, 2, . . . where Λ _s are arbitrarily chosen events in L s , respectively.

Definition 2. ([15]) A complex uncertain variable is a measurable function ζ from an uncertainty space ( Γ , L , M ) to the set of complex numbers, that is, the set { ζ ∈ B } = { γ ∈ Γ : ζ ( γ ) ∈ B }(2.2) is an event for any Borel set B of complex numbers.

When the range is the set of real numbers, we call it as an uncertain variable. Also, complex uncertain sequences are sequence of complex uncertain variables indexed by integers.

The notion of statistical convergence depends on the density of the subsets of the set ℕ of natural numbers. The following two definitions are well known (see [8, 9]).

If A is a subset of ℕ , then A _n denotes the set {k∈ A : k ≤ n } and |A_n| denotes the cardinality of A _n  . The natural density of A given by δ ( A ) = lim n → ∞ 1 n | A n | , i.e. δ ( A ) = lim n → ∞ 1 n | { k ∈ A : k ≤ n } | .

It is said that a sequence ( x k ) k ∈ ℕ is statistically convergent to x, which provided that δ ( { k ∈ ℕ : | x k − x | ≥ ε } ) = 0 , for every ɛ > 0 . If ( x k ) k ∈ ℕ is statistically convergent toxand we can write st-lim x _k  = x.

Let the increasing integer sequence k₀ = 0 and also h _s  : = k _s  - k_s-1→ ∞ as s→ ∞ such that θ ={ k _s  } be a lacunary sequence. In this work, the intervals specified by θ will be showed by I _s  : = (k _s , k_s-1] . Let q _s be a short representation of the ratio k s k s - 1 .

Definition 3. ([10]) Let θ be a lacunary sequence; the number sequence x is S _θ -convergent to ℓ provided that for every ɛ > 0, lim r 1 h r | { k ∈ I r : | x k − ℓ | ≥ ε } | = 0. Herefrom, we write S _θ  - lim x =ℓ or x _k  → ℓ (S _θ ). We describe S θ : = { x : for some ℓ , S θ - lim x = ℓ } .

On the contrary, Kostyrko et al. [13] introduced I -convergence in a metric space. This definition depends on the definition of an ideal I in ℕ .

A family I ⊂ 2 N is said to be an ideal in ℕ provided: ∅ ∈ I ; P , R ∈ I implies P ∪ R ∈ I ; P ∈ I , R ⊂ P implies R ∈ I .

A non empty family F ⊂ 2 N is said to be a filter in ℕ provided: ∅ ∉ F ; for every P , R ∈ F , P ∩ R ∈ F ; P ∈ F , P ⊂ R implies R ∈ F . Let Y ≠ ∅ . An ideal I is said to be non-trivial if I ≠ ∅ and Y ∉ I . F ( I ) = { N ∖ P : P ∈ I } is a filter on Y if and only if the I ⊂ 2 Y is a non-trivial ideal. I ⊂ 2 Y is a non-trivial ideal, that is named admissible if and only if I ⊃ { { y } : y ∈ Y } .

Definition 4. ([13]) Let I be an ideal on ℕ . The real number sequence x = (x _n ) is said to be I -convergent to L if for each ɛ > 0, the set A ( ε ) = { n ∈ ℕ : | x n − L | ≥ ε } belongs to I . If x = (x _n ) is I -convergent to L then we write I -lim x = L .

More information about I -convergent can be found from [4, 27]. Utilizing the I -convergence and statistical convergence, Das et al. [2] introduced the I -statistical convergence as follows:

Definition 5. ([2]) If for each ɛ > 0 and δ > 0 { n ∈ ℕ : 1 n | { k ≤ n : ‖ x k − ℒ ‖ ≥ ε } | ≥ δ } ∈ ℐ , a sequence (x _n ) is called to be I -statistically convergent to L .

We now recall that the concept of statistical convergence for triple sequences was presented by Şahiner et al. [23] as follows:

A function x : ℕ × ℕ × ℕ → ℝ (or ℂ ) is called a real (or complex) triple sequence. A triple sequence (x _jkl ) is said to be convergent to L in Pringsheim’s sense if for every ɛ > 0, there exists n 0 ( ɛ ) ∈ ℕ such that | x j k l − L | < ε whenever j, k, l ≥ n₀ .

Definition 6. ([23]) If δ 3 ( K ) = P − lim n , l , k → ∞ | K n l k | n l k exists, a subset K of ℕ × ℕ × ℕ is said to have natural density δ₃ (K) where the vertical bars indicate the number of (n, l, k) in K so that p ≤ n, q ≤ l, r ≤ k. When for all ɛ > 0, δ 3 ( { ( n , l , k ) ∈ ℕ × ℕ × ℕ : | x n l k − L | ≥ ε } ) = 0 , a real triple sequence x = (x _nlk ) is called to be statistically convergent to L in Pringsheim’s sense.

Throughout the paper we consider that I is the ideals of 2 ℕ ; I 2 is the ideals of 2 ℕ × ℕ and I 3 is the ideals of 2 ℕ × ℕ × ℕ .

Definition 7. ([25]) A real triple sequence (x _nlk ) is called to be I -convergent to L if for every ɛ > 0, { ( n , l , k ) ∈ ℕ × ℕ × ℕ : | x n l k − L | ≥ ε } ∈ ℐ 3 . In this case, one writes I 3 -lim x _nlk  = L .

Tripathy and Nath [26] and Kişi and Ünal [12] introduced the concepts of statistical convergence and lacunary statistical convergence for complex uncertain sequence. In this section, we present the notion of complex uncertain triple sequences and study lacunary I -statistical convergence therein. Lacunary I -statistical convergence with respect to all four aspects in uncertain space, i.e., lacunary I -statistical convergence in mean, measure, distribution and almost surely, are initiated and interrelationships among them are established.

Definition 8. ([6]) If there exist three increasing sequences of integers such that j 0 = 0 , h r = j r - j r - 1 → ∞ as r → ∞ , k 0 = 0 , h s = k s - k s - 1 → ∞ as s → ∞ , and l 0 = 0 , h t = l t - l t - 1 → ∞ as t → ∞ , the triple sequence θ_r,s,t ={ (j _r , k _s , l _t ) } is called triple lacunary sequence. Let k_r,s,t = j _r k _s l _t , h_r,s,t = h _r h _s h _t and θ_r,s,t is identified by I r , s , t = { ( j , k , l ) : j r - 1 < j ≤ j r , k s - 1 < k ≤ k s and l t - 1 < l ≤ l t } , q r = j r j r - 1 , q s = k s k s - 1 , q t = l t l t - 1 and q r , s , t = q r q s q t . Let D ⊂ ℕ × ℕ × ℕ . The number δ 3 θ ( D ) = lim r , s , t 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : ( j , k , l ) ∈ D } | expressed as θ_r,s,t-density of D, if the limit exists.

Definition 9. The complex uncertain triple sequence {ζ _rst } is called to be lacunary I -statistically convergent almost surely (a.s.) to ζ if for all ɛ, ϱ > 0 there exists an event Λ with M ( Λ ) = 1 such that { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : ‖ ζ jkl ( γ ) - ζ ( γ ) ‖ ≥ ɛ } | ≥ ϱ } ∈ I 3 , for every γ ∈ Λ . Hence we can write ζ _rst  → ζ ( S θ r , s , t ( I 3 ) a.s.).

Definition 10. The complex uncertain triple sequence {ζ _rst } is called to be lacunary I -statistically convergent in measure to ζ if { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ζ jkl - ζ ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } ∈ I 3 , for every ɛ, δ, ϱ > 0 .

Definition 11. The complex uncertain triple sequence {ζ _rst } is called to be lacunary I -statistically convergent in mean to ζ if { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : E ( ‖ ζ jkl - ζ ‖ ) ≥ ɛ } | ≥ ϱ } ∈ I 3 , for every ɛ, ϱ > 0 .

Definition 12. Let Φ, Φ _jkl be the complex uncertainty distribution of complex uncertain variables ζ, ζ _jkl respectively, where j , k , l ∈ ℕ . Then the complex uncertain triple sequence {ζ _rst } is called to be lacunary I -statistically convergent in distribution to ζ if for all ɛ, ϱ > 0, { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : ‖ Φ jkl ( z ) - Φ ( z ) ‖ ≥ ɛ } | ≥ ϱ } ∈ I 3 , for all complex z at which Φ (z) is continuous.

Now, relationships between lacunary I -statistical convergence a.s., lacunary I -statistical convergence in mean, lacunary I -statistical convergence in measure, lacunary I -statistical convergence in distribution will be studied.

Theorem 1. If the complex uncertain triple sequence {ζ _rst } lacunary I -statistically converges in the mean to ζ, then {ζ _rst } lacunary I -statistically converges to ζ in measure.

Proof. Let the complex uncertain triple sequence {ζ _rst } be lacunary I -statistically convergent in mean to ζ . For any taken ɛ, ϱ, δ > 0 with help of the Markov inequality, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ζ jkl - ζ ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } ⊆ { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : ( E ( ‖ ζ jkl - ζ ‖ ) ɛ ) ≥ ϱ } | ≥ δ } ∈ I 3 . Hence {ζ _rst } lacunary I -statistically converges in measure to ζ . □

Converse of above theorem is not true in general, i.e. lacunary I -statistically convergence in measure does not imply lacunary I -statistically convergence in mean. This can be demonstrated by the example given below.

Example 1. Take into account the uncertainty space ( Γ , L , M ) . It becomes Γ ={ γ₁, γ₂, . . . } with l M { Λ } = { sup γ r + s + t ∈ Λ 1 r + s + t + 1 , if sup γ r + s + t ∈ Λ 1 r + s + t + 1 < 0.5 , 1 - sup γ r + s + t ∈ Λ C 1 r + s + t + 1 , if sup γ r + s + t ∈ Λ C 1 r + s + t + 1 < 0.5 , 0.5 , otherwise , also ζ rst ( γ ) = { ( r + s + t + 1 ) i , if γ = γ r + s + t , 0 , otherwise , describes the complex uncertain variables for r , s , t ∈ ℕ and ζ (γ) ≡ 0, ∀γ ∈ Γ . For some small numbers ɛ, ϱ, δ > 0 and r, s, t ≥ 2, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ζ jkl - ζ ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( γ : ‖ ζ jkl ( γ ) - ζ ( γ ) ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M { γ = γ j + k + l } ≥ ϱ } | ≥ δ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : 1 j + k + l + 1 ≥ ϱ } | ≥ δ } ∈ I 3 . Thus, the complex triple sequence {ζ _rst } lacunary I -statistically converges in measure to ζ .

Φ _rst is the uncertainty distribution function for r, s, t ≥ 2 and of the complex uncertain variable ||ξ _rst  - ξ|| = ||ξ _rst || . This is, Φ rst ( x ) = { 0 , if x < 0 , ( 1 - 1 r + s + t + 1 ) , if 0 ≤ x < ( r + s + t + 1 ) , 1 , if x ≥ r + s + t + 1 . And, { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : E ( ‖ ζ jkl - ζ ‖ - 1 ) ≥ ɛ } | ≥ ϱ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : ( [ ∫ 0 j + k + l + 1 1 - ( 1 - 1 j + k + l + 1 ) dx ] - 1 ) ≥ ɛ } | ≥ ϱ } .

Hence, for each r, s, t ≥ 2, and for every ɛ, ϱ > 0, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : E ( ‖ ζ jkl - ζ ‖ ) ≥ ɛ } | ≥ ϱ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t } | ≥ ϱ } ∈ F ( I 3 ) , which is impossible. So, the complex uncertain triple sequence {ζ _rst } doesn’t lacunary I -statistically converges in mean to ζ .

Theorem 2. Let the complex uncertain triple sequence {ζ _rst } where {ξ _rst } is the real part and {η _rst } is the imaginary part, for r , s , t ∈ ℕ . When uncertain triple sequences {ξ _rst } and {η _rst } lacunary I -statistically converges to ξ as measure and γ, respectively, complex uncertain triple sequence {ζ _rst } lacunary I -statistically converges to ζ = ξ + iη as measure.

Proof. Let {ξ _rst } and {η _rst } lacunary I -statistically converges to ξ and η respectively in measure. Then for any small numbers ɛ, ϱ, δ > 0, { ( r , s , t ) : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ξ jkl - ξ ‖ ≥ ɛ 2 ) ≥ ϱ } | ≥ δ } ∈ I 3 and { ( r , s , t ) : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ η jkl - η ‖ ≥ ɛ 2 ) ≥ ϱ } | ≥ δ } ∈ I 3 . Note that ‖ ζ rst - ζ ‖ = | ξ rst - ξ | 2 + | η rst - η | 2 . Therefore, we have { ‖ ζ rst - ζ ‖ ≥ ɛ } ⊂ { ‖ ξ rst - ξ ‖ ≥ ɛ 2 ∪ ‖ η rst - η ‖ ≥ ɛ 2 } . Taking advantage of the subadditivity axiom of uncertain measure, we get { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ζ jkl - ζ ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } ⊆ { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ξ jkl - ξ ‖ ≥ ɛ 2 ) ≥ ϱ } | ≥ δ } ∪ { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ η jkl - η ‖ ≥ ɛ 2 ) ≥ ϱ } | ≥ δ } ∈ I 3 . Hence, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ζ jkl - ζ ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } ∈ I 3 . That is, {ζ _rst } lacunary I -statistically converges in measure to ζ . □

To proof of Theorem 3, we give the following definition:

Definition 13. ([11]) (i) There exists a t ∈ ℝ so that M t ∉ I 3 , then we have I 3 - lim sup x = sup { t ∈ ℝ : M t ∉ I 3 }(3.1) where M t = { ( n , l , k ) ∈ ℕ × ℕ × ℕ : 1 nkl | { p ≤ n , q ≤ l , r ≤ k : x pqr > t } | > γ } .

When M t ∈ I 3 takes for each one t ∈ ℝ , we have I 3 - lim sup x = - ∞ .

(ii) There exists a t ∈ ℝ so that M t ∉ I 3 then we have I 3 - lim inf x = inf { t ∈ ℝ : M t ∉ I 3 }(3.2) where M t = { ( n , l , k ) ∈ ℕ × ℕ × ℕ : 1 nkl | { p ≤ n , q ≤ l , r ≤ k : x pqr < t } | > γ } .

When M t ∈ I 3 takes for each one t ∈ ℝ , we have I 3 - lim inf x = + ∞ .

Theorem 3. Let complex uncertain triple sequence {ζ _rst } where {ξ _rst } is the real part and {η _rst } is the imaginary part, for r , s , t ∈ ℕ . When uncertain triple sequences {ξ _rst } and {η _rst } lacunary I -statistically converges to ξ as measure and η, respectively, complex uncertain triple sequence {ζ _rst } lacunary I -statistically converges to ζ = ξ + iη as distribution.

Proof. The complex uncertainty distribution Φ should have a definite point of continuity z = u + iv. Otherwise, we have { ξ rst ≤ u , η rst ≤ v } = { ξ rst ≤ u , η rst ≤ v , ξ ≤ x ˆ , η ≤ y ˆ } ∪ { ξ rst ≤ u , η rst ≤ v , ξ > x ˆ , η > y ˆ } ∪ { ξ rst ≤ u , η rst ≤ v , ξ ≤ x ˆ , η > y ˆ } ∪ { ξ rst ≤ u , η rst ≤ v , ξ > x ˆ , η ≤ y ˆ } ⊂ { ξ ≤ u , η ≤ v } ∪ { | ξ rst - ξ | ≥ x ˆ - u } ∪ { | η rst - η | ≥ y ˆ - v } for any x ˆ > u , y ˆ > v . From here with the axiom of subadditivity, Φ rst ( z ) = Φ rst ( u + iv ) ≤ Φ ( x ˆ + i y ˆ ) + M { | ξ rst - ξ | ≥ x ˆ - u } + M { | η rst - η | ≥ y ˆ - v } . Since {ξ _rst } and {η _rst } lacunary I -statistically converge to ξ as measure and η, respectively, hence, for every small numbers ɛ, ϱ > 0, we get { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ξ jkl - ξ ‖ ≥ x ˆ - u ) ≥ ɛ } | ≥ ϱ } ∈ I 3 and { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ η jkl - η ‖ ≥ y ˆ - v ) ≥ ɛ } | ≥ ϱ } ∈ I 3 . Thus, we obtain I 3 - lim sup r , s , t → ∞ Φ r , s , t ( z ) ≤ Φ ( x ˆ + i y ˆ ) for any x ˆ > u , y ˆ > v . Letting x ˆ + i y ˆ → u + iv , we get I 3 - lim sup r , s , t → ∞ Φ r , s , t ( z ) ≤ Φ ( z ) .(3.3) Furthermore, we have { ξ ≤ a , η ≤ b } = { ξ rst ≤ u , η rst ≤ v , ξ ≤ a , η ≤ b } ∪ { ξ rst ≤ u , η rst > v , ξ ≤ a , η ≤ b } ∪ { ξ rst > u , η rst ≤ v , ξ ≤ a , η ≤ b } ∪ { ξ rst > u , η rst > v , ξ ≤ a , η ≤ b } ⊂ { ξ rst ≤ u , η rst ≤ v } ∪ { | ξ rst - ξ | ≥ u - a } ∪ { | η rst - η | ≥ v - b } for any a < u, b < v. This means, Φ ( a + ib ) ≤ Φ rst ( u + iv ) + M { ‖ ξ rst - ξ ‖ ≥ u - a } + M { ‖ η rst - η ‖ ≥ v - b } . Inasmuch as { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ζ jkl - ζ ‖ ≥ u - a ) ≥ ɛ } | ≥ ϱ } ∈ I 3 and { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ η jkl - η ‖ ≥ v - b ) ≥ ɛ } | ≥ ϱ } ∈ I 3 we gain Φ ( a + ib ) ≤ I 3 - lim inf r , s , t → ∞ Φ rst ( u + iv ) for any a < u, b < v . Taking a + ib → u + iv, we get Φ ( z ) ≤ I 3 - lim inf r , s , t → ∞ Φ rst ( z )(3.4) From (3.3) and (3.4) we find Φ _rst  (z) → Φ (z) as r, s, t → ∞ . This is the complex uncertain triple sequence {ζ _rst } and it is lacunary I -statistically convergent in distribution to ζ = ξ + iη . □

Remark 1. Lacunary I -statistically convergence in distribution doesn’t allude to the lacunary I -statistically convergence in measure. This indicates that the contrary of the above theorem isn’t necessarily true.

The following example shows the above result.

Example 2. For this Γ ={ γ₁, γ₂, γ₃ } case, there are 8 (eight) events. Define M { γ 1 } = 0.6 , M { γ 1 , γ 2 } = 0.8 , M { ∅ } = 0 , M { γ 2 } = 0.3 , M { γ 1 , γ 3 } = 0.7 , M { Γ } = 1 . M { γ 3 } = 0.2 , M { γ 2 , γ 3 } = 0.4 , Obviously, the set function M is an uncertain measure. We describe the complex uncertain variables as ζ rst ( γ ) = { i , if γ = γ 1 , - i , if γ = γ 2 , 2 i , if γ = γ 3 .(3.5) We describe -ζ _rst  = ζ for r , s , t ∈ ℕ . Therefore, by Example 1.1 and 1.5 in [15] we can say {ζ _rst } lacunary I -statistically converges in distribution to ζ . But, the complex uncertain triple sequence {ζ _rst } doesn’t lacunary I -statistically convergence in measure to ζ .

Lacunary I -statistically convergence almost surely doesn’t indicate lacunary I -statistically convergence in measure.

Example 3. With Borel algebra and Lebesque measure, take the uncertainty space ( Γ , L , M ) as [0, 1]. There are integers y₁, y₂ and y₃ such that r = 2 ^{y ₁}  + p, s = 2 ^{y ₂}  + p and t = 2 y 3 + p ∈ ℕ , for any positive integer r, s, t where p is an integer between 0 and min { 2 ^{y ₁} , 2 ^{y ₂} , 2 ^{y ₃}  } - 1 . If so, we determine a complex uncertain variable by ζ rst ( γ ) = { i , if p 2 y 1 + y 2 + y 3 ≤ γ ≤ p + 1 2 y 1 + y 2 + y 3 , 0 , otherwise ,(3.6) for r , s , t ∈ ℕ and ζ ≡ 0 . For some small numbers ɛ, ϱ, δ > 0 and r, s, t ≥ 2, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( ‖ ξ jkl - ξ ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M ( γ : ‖ ζ jkl ( γ ) - ζ ( γ ) ‖ ≥ ɛ ) ≥ ϱ } | ≥ δ } = { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : M { γ jkl } ≥ ϱ } | ≥ δ } ∈ I 3 , . Thus, the triple sequence {ζ _rst } lacunary I -statistically converges to ζ as measure. Also, for all ɛ, ϱ > 0, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : E ( ‖ ζ jkl - ζ ‖ ) ≥ ɛ } | ≥ ϱ } ∈ I 3 . Hence, the sequence {ζ _rst } also lacunary I -statistically converges in mean to ζ . But, there exists an infinite number of intervals of the form [ p 2 y 1 + y 2 + y 3 , p + 1 2 y 1 + y 2 + y 3 ] containing γ, for any γ ∈ [0, 1] . Thus, ζ _rst  (γ) doesn’t lacunary I -statistically converge to 0 . Another way, the triple uncertain sequence {ζ _rst } doesn’t lacunary I -statistically converge a.s. to ζ . This completes the proof.

Lacunary I -statistically convergence a.s. doesn’t allude to the lacunary I -statistically convergence in mean.

Example 4. Take into account the uncertainty space ( Γ , L , M ) to be Γ ={ γ₁, γ₂, γ₃, . . . } with M { Λ } = ∑ γ d , γ e , γ f ∈ Λ 1 2 d + e + f .(3.7) For r , s , t ∈ ℕ and ζ (γ) ≡ 0, ∀γ ∈ Γ, ζ rst ( γ ) = { i 2 r + s + t , if γ = γ r + s + t , 0 , otherwise ,(3.8) is the complex uncertain variables. Thereafter, {ζ _rst } sequence, lacunary I -statistically converges a.s. to ζ . But, the uncertainty distributions of ||ζ _rst || are Φ rst ( x ) = { 0 , if x < 0 , 1 - 1 2 r + s + t , if 0 ≤ x < 2 r + s + t , 1 if x ≥ 2 r + s + t , for r , s , t ∈ ℕ , respectively. Then, we have { ( r , s , t ) ∈ ℕ × ℕ × ℕ : 1 h r , s , t | { ( j , k , l ) ∈ I r , s , t : E ( ‖ ζ jkl - ζ ‖ ) ≥ 1 } | ≥ ϱ } ∈ I 3 . Therefore, the complex uncertain triple sequence {ζ _rst } doesn’t lacunary I -statistically converge in mean to ζ .

From the example given above, we can acquire that lacunary I -statistically convergence in mean doesn’t allude to the lacunary I -statistically converge a.s.

Example 5. Take into account the uncertainty space ( Γ , L , M ) to be Γ ={ γ₁, γ₂, γ₃, γ₄ } with M { Λ } = { 0 , if Λ = ∅ 0.6 , if γ 1 ∈ Λ 0.4 , if γ 1 ∉ Λ 1 , if Λ = Γ .(3.9) For r , s , t ∈ ℕ and ζ (γ) ≡ 0, ∀γ ∈ Γ, ζ rst ( γ ) = { i , if γ = γ 1 2 i , if γ = γ 2 3 i , if γ = γ 3 4 i , if γ = γ 4 0 , otherwise(3.10) is the complex uncertain variables. We find that the {ζ _rst } sequence is lacunary I -statistically convergent to ζ with respect to almost surely, but the {ζ _rst } sequence is not lacunary I -statistically convergent in the measure.

If the complex uncertainty distribution of the complex uncertain variables ζ _rst and ζ are Φ _rst  (z) and Φ (z) respectively in the uncertainty space taken in the above example, then for r , s , t ∈ ℕ Φ rst ( z ) = { 0 , if a < 0 , - ∞ < b < ∞ 0 , if a ≥ 0 , b < 1 0.6 , if a ≥ 0 , 1 ≤ b < 4 1 if a ≥ 0 , b ≥ 4(3.11) and Φ ( z ) = { 0 , if a < 0 , - ∞ < b < ∞ 0 , if a ≥ 0 , b < 0 1 if a ≥ 0 , b ≥ 0 .(3.12) From the examples given above, it is obvious that the complex uncertain triple sequence {ζ _rst } isn’t lacunary I -statistically converge in distribution to ζ .

Here, the study of lacunary ideal statistical convergence in mean, in measure, in distribution, with respect to almost surely of a complex uncertain triple sequence has been made and interrelationships among them were established. The results of the paper are expected to be a source for researchers in the areas of convergence methods for triple sequences and applications in uncertainty and summability theory. Also, these concepts can be generalized and applied for further studies. For example, this study can be extended by introducing I -statistically Cauchy triple sequence of complex uncertain variables. In future studies on this topic, the ideal invariant convergence by using triple sequences can be defined and examined for complex uncertain sequence.