Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence

Abstract

We explore the realm of uncertainty theory by investigating diverse notions of convergence and statistical convergence concerning complex uncertain sequences. Complex uncertain variables can be described as measurable functions mapping from an uncertainty space to the set of complex numbers. They are employed to represent and model complex uncertain quantities. We introduce the concept of lacunary almost statistical convergence of order $\alpha$$(0<\alpha \le 1)$ for complex uncertain sequences, examining various aspects of uncertainty such as distribution, mean, measure, uniformly almost sure convergence and almost sure convergence. Additionally, we establish connections between the constructed sequence spaces by providing illustrative instances. Importantly, lacunary almost statistical convergence provides a flexible framework for handling sequences with irregular behavior, which often arise in uncertain environments with imprecise data. This makes our approach particularly useful in practical fields such as engineering, data modeling and decision-making, where traditional deterministic methods are not always applicable. Our approach offers a more flexible and realistic framework for approximating functions in uncertain environments where classical convergence may not apply. Thus, this study contributes to approximation theory by extending its tools to settings involving imprecise or noisy data.

1. Introduction

In real life, people are often faced with situations that require them to make decisions in a state of uncertainty. According to Liu [1], there are two ways to deal with indeterminacy. Probability theory is the most well-known theory and is inevitable in situations when the frequencies are extremely close to the distribution function. On the other hand, when there is no sample available to approximate a probability distribution, probability theory never permits exact solutions. For instance, we are unable to determine the strength of the existing bridge. Consequently, finding sufficient sample data for some events may be challenging. In order to determine the probability that each event would occur under these conditions, we must rely on the domain experts. In order to address these types of situations, Liu [2] looked into the uncertainty theory idea. While probability theory is based on the frequency of random events, uncertainty theory is based on the degree of confidence. We apply the principle of uncertainty to effectively address both fuzziness and randomness. In recent times, an aspect of mathematics referring to uncertainty theory has been used to model human uncertainty. Liu proposed a revised version of this speculative idea [3]. Furthermore, Liu used this theory of uncertainty in sequences and showed convergence in mean, distribution, measure and almost sure convergence to prove the property of convergence of uncertain measures. By widening the idea of real uncertain variables into complex uncertain variables, Peng [4] provided uncertainty theory a new approach, which Chen et al. [5] further developed. Datta and Tripathy [6] made further generalizations to their approach by introducing double sequences of complex uncertain variables. After that, You [7] mentioned the uniformly almost sure convergence of complex uncertain sequences in 2009. Classical convergence demands all terms of a sequence to closely approach a limit, and statistical convergence allows for occasional deviations—making it ideal for uncertain data environments. By integrating statistical convergence with uncertainty theory, we can better understand the long-term behavior of complex uncertain sequences, which is essential in many practical applications—such as engineering, data modeling and control systems. It is often necessary to approximate complex or uncertain functions when exact forms are unknown. When dealing with noisy or missing sensor data in engineering systems, complex uncertain sequences provide reliable means to model such uncertainties. The concept of lacunary almost statistical convergence ensures accurate approximation even if some data points stray from expected values. Nonetheless, applying this method requires careful selection of sequence gaps and uncertainty profiles, which can be difficult with limited data. Practical application is strengthened by incorporating expert knowledge and dynamic adjustment techniques. Classical methods of approximation rely heavily on deterministic models and well-defined convergence. However, in environments filled with uncertainty, these traditional methods may fall short. This is where the combination of uncertainty theory and statistical convergence becomes especially powerful. Numerous mathematicians have contributed to this field, examining its properties and extending its scope; see ([8,9,10]) and many more. By studying how uncertain sequences behave and converge, we gain new tools for constructing and evaluating approximations in situations where classical techniques cannot be applied reliably. This work aims to contribute to this growing area by extending the idea of convergence to complex uncertain variables and exploring its implications for function approximation in uncertain environments. Throughout the paper, we take $\mathbb{R}$ and $\mathbb{N}$ as real numbers and natural numbers, respectively.

2. Preliminaries

The notion of statistical convergence, a generalization of the classical notion of convergence, was first introduced by Zygmund in the inaugural edition of his celebrated monograph, published in Warsaw in 1935 [11] and independently by Steinhaus [12] and Fast [13]. This concept was later revisited independently by Schoenberg [14]. Unlike classical convergence, which requires that all but finitely many terms of a sequence lie within an arbitrarily small neighborhood of the limit, statistical convergence relaxes this requirement. Instead, it considers a sequence convergent if the density of terms satisfying the convergence criterion approaches unity. This broader framework enables the study of sequences beyond the constraints of traditional convergence.

Every convergent sequence is statistically convergent. But the converse need not be true in general.

Example 1.

Define a sequence $x=\{x_k\}$ by

$$x_k=\left\{\begin{array}{cc}1,\hfill & ifk=n^{2}forn\in \mathbb{N}\hfill \\ \hfill & \\ 0,\hfill & otherwise\hfill \end{array}\right.$$

i.e., x=$\{1,0,0,1,0,0,0,0,1,0,0,0,\dots .\}$.

For this we first observe that $|\{k\le n:x_k\ne 0\}|\le \sqrt{n}$ (since $x_k$ is non-zero only when k is a square).

Therefore, ${\displaystyle \underset{n\to \infty }{\lim }\frac{1}{n}|k\le n:x_k\ne 0|\le \underset{n\to \infty }{\lim }\frac{\sqrt{n}}{n}=\underset{n\to \infty }{\lim }\frac{1}{\sqrt{n}}=0.}$ Thus, the sequence $\{x_k\}$ is statistically convergent, but it is evident that it is not convergent as it is finitely oscillating.

A real sequence $(z_m)$ is said to be statistically convergent to a real number z if for a given $\epsilon$> 0,

$${\displaystyle \underset{\begin{array}{c}\hfill (n\to \infty )\end{array}}{\lim }\frac{1}{n}|\{m\le n:|z_m-z|\ge \epsilon \}|=0.}$$

Later, Tripathy and Nath [10] used complex uncertain sequences to study statistical convergence.

Now, we present some definitions of uncertainty theory given by (Liu [1]).

Definition 1.

Consider ℓ to be a $\sigma -$algebra on a nonempty set Ω. A mapping $\widehat{\mathit{M}}:\ell \to [0,1]$ is called an uncertain measure if it satisfies

(i) 

$\widehat{\mathit{M}}(\mathrm{\Omega })=1$;

(ii) 

$\widehat{\mathit{M}}\{\mathrm{\Lambda }\}+\begin{array}{c}\hfill \widehat{\mathit{M}}\{{\mathrm{\Lambda }}^c\}\end{array}=1$;

(iii) 

For each countable sequence of $\{{\mathrm{\Lambda }}_j\}\in \ell ,$

$$\begin{array}{ccc}\hfill \widehat{\mathit{M}}\left\{{\displaystyle \bigcup _{j=1}^{\infty }}{\mathrm{\Lambda }}_j\right\}& \le & {\displaystyle {\displaystyle \sum _{j=1}^{\infty }}\widehat{\mathit{M}}\{{\mathrm{\Lambda }}_j\}}\hfill \end{array}$$

where each element $\mathrm{\Lambda }\in \ell$ is called an event. The triplet $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ is known as uncertainty space.

Liu [15] discovered the product criteria of an uncertain measure for compound events as follows:

(iv) 

Let $({\mathrm{\Omega }}_j,{\ell }_j,{\widehat{\mathit{M}}}_j)$ be an uncertainty space where $j=1,2,3,\begin{array}{c}\hfill \dots \end{array}.$ The product uncertain measure $\widehat{\mathit{M}}$ is a measure satisfying

$$\begin{array}{ccc}\hfill \widehat{\mathit{M}}\left\{{\displaystyle \prod _{j=1}^{\infty }}{\mathrm{\Lambda }}_j\right\}& =& {\displaystyle \underset{j=1}{\overset{\infty }{\bigwedge }}}{\widehat{\mathit{M}}}_j\{{\mathrm{\Lambda }}_j\},\hfill \end{array}$$

for any ${\mathrm{\Lambda }}_j$∈${\ell }_j.$

Definition 2.

The uncertain variable z is a measurable function from $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set

$$\{z\in B\}=\{\omega \in \mathrm{\Omega }:z(\omega )\in B\}$$

is an event.

Definition 3.

The uncertainty distribution ϕ of an uncertain variable z is defined as

$$\begin{array}{ccc}\hfill \varphi (x)& =& \widehat{\mathit{M}}\{z\le x\},\forall real numberx.\hfill \end{array}$$

Definition 4.

The expected value E of an uncertain variable z is defined as

$$\begin{array}{ccc}\hfill E\left[z\right]& =& {\int }_0^{+\infty }\widehat{\mathit{M}}\{z\ge v\}dv-{\int }_{-\infty }^0\widehat{\mathit{M}}\{z\le v\}dv\hfill \end{array}$$

only if one of the two integrals is finite.

Chen et al. [5] presented a notion in complex uncertain sequences, which is shown below.

Definition 5.

The complex uncertain sequence $({\zeta }_j)$ is said to be almost surely convergent to ζ if ∃Λ with $\widehat{\mathit{M}}(\mathrm{\Lambda })=1$ such that

$$\begin{array}{c}\hfill \underset{j\to \infty }{\lim }\parallel {\zeta }_j(\omega )-\zeta (\omega )\parallel =0,\end{array}$$

for every $\omega \in \mathrm{\Lambda }.$

Definition 6.

The complex uncertain sequence $({\zeta }_j)$ is said to be convergent in measure to ζ if

$$\begin{array}{c}\hfill \underset{j\to \infty }{\lim }\widehat{\mathit{M}}\{\parallel {\zeta }_j-\zeta \parallel \ge \epsilon \}=0,\end{array}$$

∀ε $>0.$

Definition 7.

The complex uncertain sequence $({\zeta }_j)$ is said to be convergent in mean to ζ if

$$\begin{array}{c}\hfill \underset{j\to \infty }{\lim }\mathbf{E}[\parallel {\zeta }_j-\zeta \parallel ]=0.\end{array}$$

Definition 8.

If $\varphi ,{\varphi }_1,{\varphi }_2,\dots$ are uncertainty distributions of uncertain variables $\zeta ,{\zeta }_1,{\zeta }_2,\dots$, respectively, then a complex uncertain sequence $({\zeta }_j)$ is convergent in distribution to ζ if

$$\begin{array}{ccc}\hfill \underset{j\to \infty }{\lim }{\varphi }_j(v)& =& \varphi (v),\hfill \end{array}$$

∀$v\in \mathbb{R},$at which $\varphi (v)$ is continuous.

Definition 9.

The complex uncertain sequence $({\zeta }_j)$ is uniformly almost surely convergent to ζ if ∃ a sequence of events $(E^{\prime }(j))$ with $\widehat{\mathit{M}}(E^{\prime }(j))$ approaching zero such that $({\zeta }_j)$ converges uniformly to ζ in $\mathrm{\Omega }-E^{\prime }(j),$ for fixed $j\in \mathbb{N}.$

Remark 1.

Suppose ${\zeta }_j={\varrho }_j+i{\varsigma }_j,$ for $j\in \mathbb{N}$ and $\zeta =\varrho +i\varsigma ;$ we get

$$\parallel {\zeta }_j-\zeta \parallel =\sqrt{{({\varrho }_j-\varrho )}^2+{({\varsigma }_j-\varsigma )}^2}$$

as an uncertain variable.

The concept of lacunary sequences was introduced by Freedman et al. [16]. A lacunary sequence is an increasing non-negative integer sequence $\theta =(j_r)$ with $j_0=0$ and $h_r=j_r-j_{r-1}\to \infty$ as $r\to \infty$. Throughout the paper, the intervals determined by r will be denoted by $I_r=(j_{r-1},j_r]$, and later, we refer to Fridy and Orhan [17] to introduce the concept of lacunary statistical convergence. A real sequence $z=(z_j)$ is said to be lacunary statistically convergent to a number L if for a given $\epsilon >0,$

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{h_r}}|\{j\in I_r: |z_j-L|\ge \epsilon \}|=0.$$

Kişi and Ünal [9] studied lacunary statistical convergence for complex uncertain sequences.

Almost convergence was first defined by Lorentz [18] in 1948 using the idea of a Banach limit. A bounded sequence $\{x_l\}$ is known to be almost convergent to a number z if and only if

$${\displaystyle \underset{j\to \infty }{\lim }\frac{1}{j+1}{\displaystyle \sum _{l=0}^j}{x}_{k+l}=z}$$

uniformly in k for some z ∈ $\mathbb{C}$.

For fundamental theorems in functional analysis and summability theory, readers may refer to recent textbooks [19,20], while further insights on almost convergence and related topics can be found in [21,22,23,24]. Savaş [25], by using fuzzy numbers, introduced the concept of “almost convergence”, which was further generalized by Saha, Tripathy and Roy [26] by introducing almost convergence of complex uncertain sequences. In 2021, Das et al. introduced the concept of almost convergence for complex uncertain sequences in both double [27] and triple sequences [6]. Following this, Raj et al. [28] introduced the concept of almost $\lambda$-statistical convergence for complex uncertain sequences. In 2023, Nath et al. [29] proposed the concept of strongly almost convergence in sequences of complex uncertain variables. More recently, Das et al. [30] established a relationship between convergence and almost convergence of complex uncertain sequences. Motivated from these works, we introduce the concept of lacunary almost statistical convergence of order $\alpha$ for complex uncertain sequences via measure, mean, distribution, almost sure convergence and uniformly almost sure convergence and establish some relationships among these notions. Furthermore, when the converse of a theorem does not hold, a counterexample is provided to support the result.

3. Main Results

In this section, we present the relationship between complex uncertain sequences by using lacunary almost statistical convergence. (Figure of Interrelationships can be seen in Figure 1):

Definition 10.

The complex uncertain sequence $\left\{{\zeta }_j\right\}$ is known to be lacunary almost statistically convergent of order α in concert with almost sure convergence to ζ if $\forall \epsilon >0$ there exists an event Λ with $\widehat{\mathit{M}}$(Λ) = 1 such that

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\Vert f_j{,}_k(\omega )-\zeta (\omega )\Vert \ge \epsilon \left\}\right|=0,$$

uniformly in k $>0$ and ∀$\omega \in \mathrm{\Lambda },$ where Λ represents an uncertain event, $\widehat{\mathit{M}}$(Λ) = 1 and ${\displaystyle f_j{,}_k(\omega )=\frac{{\zeta }_k+{\zeta }_{k+1}+\begin{array}{c}\hfill \dots \end{array}+{\zeta }_{k+j}}{j+1}=\frac{1}{j+1}{\displaystyle \sum _{l=0}^j}{\zeta }_{k+l}(\omega )}$.

Definition 11.

The complex uncertain sequence $\{{\zeta }_j\}$ is known to be lacunary almost statistically convergent of order α in measure to ζ if ∀ε, $\delta >0$,

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}\{\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon \}\ge \delta \left\}\right|=0.$$

Definition 12.

The complex uncertain sequence $\{{\zeta }_j\}$ is known to be lacunary almost statistically convergent of order α in mean to ζ if ∀, ε$>0$

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\mathbf{E}\{\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon \}\left\}\right|=0.$$

Definition 13.

If the $\varphi ,{\varphi }_1{,}_k,{\varphi }_2{,}_k,\dots$ are uncertainty distributions of uncertain variables $\zeta ,{\zeta }_1,{\zeta }_2,\dots$, respectively, then the complex uncertain sequence $\{{\zeta }_j\}$ lacunary almost statistically converges in distribution to ζ if for every ε $>0$,

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}\{\parallel {\varphi }_j{,}_k\begin{array}{c}\hfill (v)\end{array}-\varphi \begin{array}{c}\hfill (v)\end{array}\parallel \ge \epsilon \}\left\}\right|=0,$$

for all v = a+ib at which ϕ$(v)$ is continuous.

Definition 14.

The complex uncertain sequence $\{{\zeta }_j\}$ is known to be lacunary almost statistically convergent of order α and uniformly almost surely convergent to ζ if for every ε$>0$∃$\delta >0$and a sequence of events $\{E_j^{\prime }\}$ such that

$$\begin{array}{cc}& \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:|\widehat{\mathit{M}}(E_j^{\prime })-0|\ge \epsilon \left\}\right|=0,\hfill \\ \hfill \Rightarrow & \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:|f_j{,}_k(\omega )-\zeta (\omega )|\ge \delta \left\}\right|=0,\hfill \end{array}$$

where $E_j^{\prime }$ represents a sequence of events.

Lemma 1.

Let $\xi$ be a non-negative uncertain variable (i.e., $\xi (\omega )\ge 0$ for all $\omega \in \mathrm{\Omega }$), and let $a>0$ be any positive number. Then, the uncertain measure of the set where $\xi (\omega )$ is at least a satisfies

$$\widehat{M}(\{\omega \in \mathrm{\Omega }:\xi (\omega )\ge a\})\le {\displaystyle \frac{E\left[\xi \right]}{a}}.$$

Proof. 

Define the set $A=\{\omega \in \mathrm{\Omega }:\xi (\omega )\ge a\}$, and let ${\chi }_A(\omega )$ denote its indicator function:

$${\chi }_A(\omega )=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\omega \in A,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Since $\xi (\omega )\ge a$ for all $\omega \in A$, we have

$$a·{\chi }_A(\omega )\le \xi (\omega )\mathrm{for}\mathrm{all}\omega \in \mathrm{\Omega }.$$

Taking the expected value of both sides yields

$$E[a·{\chi }_A(\omega )]\le E\left[\xi (\omega )\right].$$

Using the linearity of expectation and the fact that $E\left[{\chi }_A(\omega )\right]=\widehat{M}(A)$, we get

$$a·\widehat{M}(A)\le E\left[\xi \right].$$

Dividing both sides by a gives

$$\widehat{M}(A)\le {\displaystyle \frac{E\left[\xi \right]}{a}}.$$

This completes the proof. □

Theorem 1.

If the uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent in mean to ζ, then it is also lacunary almost statistically convergent in measure to ζ. But the converse does not hold, in general.

Proof. 

Suppose the uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent in mean to $\zeta$. Then, for every $\epsilon$$>0$, we have

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\mathbf{E}\{\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon \}\left\}\right|=0.$$

Now, using Markov’s inequality, we get

$$\widehat{M}(\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon )\le {\displaystyle \frac{E\left[\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \right]}{\epsilon }}.$$

Hence,

$$\{j\in I_r:\widehat{M}(\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon )\ge \delta \}\subseteq \{j\in I_r:E[\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel ]\ge \epsilon \delta \}.$$

This implies

$${\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left(\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \right)\ge \delta \left\}\right|\le$$

$${\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:E[\parallel f_{j,k}-\zeta \parallel ]\ge \epsilon \delta \left\}\right|\to 0$$

as $r\to \infty$. Thus, the sequence is lacunary almost statistically convergent in measure to $\zeta$. This proves the result. Now, for the converse we may illustrate by the example given below. □

Example 2.

Consider the uncertainty space $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ with $\mathrm{\Omega }=\{{\omega }_1,{\omega }_2,\dots \}$ and a set function $\widehat{\mathit{M}},$ an uncertain measure satisfying the three axioms of uncertainty and having uncertain measurable functions as follows:

$$\widehat{\mathit{M}}\{\mathrm{\Lambda }\}=\left\{\begin{array}{cc}{\sup }_{{\omega }_{j\in \begin{array}{c}\hfill \mathrm{\Lambda }\end{array}}}{\displaystyle \frac{1}{(k^2+5)j}},\hfill & if{\sup }_{{\omega }_{j\in \mathrm{\Lambda }}}{\displaystyle \frac{1}{(k^2+5)j}}<0.5;\hfill \\ \\ 1-{\sup }_{{\omega }_{j\in \begin{array}{c}\hfill {\mathrm{\Lambda }}^c\end{array}}}{\displaystyle \frac{1}{(k^2+5)j}},\hfill & if{\sup }_{{\omega }_{j\in {\mathrm{\Lambda }}^c}}{\displaystyle \frac{1}{(k^2+5)j}}<0.5;\hfill \\ \\ 0.5,\hfill & otherwise.\hfill \end{array}\right.$$

Now, we define the complex uncertain variable

$${\zeta }_j(\omega )=\left\{\begin{array}{cc}(k^2+5)ji,\hfill & if\omega ={\omega }_j;\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$$

for $k=1,2,\dots$ and $\zeta \equiv 0.$ Then, for some given small number ε $>0$ and $k,j\ge 2,$ we have

$$\begin{array}{cc}& \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}(\parallel f_j{,}_k-\zeta \parallel \ge \epsilon )\ge \delta \left\}\right|\hfill \\ \hfill =& \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}(\omega :\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon )\ge \delta \left\}\right|\hfill \\ \hfill =& \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}({\omega }_j)\ge \delta \left\}\right|=0.\hfill \end{array}$$

So, the sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent in measure to ζ.

• However, for j, k ≥2 the uncertainty distribution ${\varphi }_j{,}_k$ of the uncertain variable $\parallel f_j{,}_k-\zeta \parallel$ is

$${\varphi }_j{,}_k\begin{array}{c}\hfill (v)\end{array}=\left\{\begin{array}{cc}0,\hfill & ifv<0;\hfill \\ 1-{\displaystyle \frac{1}{(k^2+5)j}},\hfill & if0\le v<(k^2+5)j;\hfill \\ 1,\hfill & ifv\ge (k^2+5)j.\hfill \end{array}\right.$$

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{\lim }& {\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\mathbf{E}(\parallel f_j{,}_k-\zeta \parallel -1)\left\}\right|=[{\int }_0^{(k^2+5)j}1-(1-\hfill & \hfill {\displaystyle \frac{1}{(k^2+5)j}})du-1]=0.\end{array}$$

Thus, the sequence $\{{\zeta }_j\}$ is not lacunary almost statistically convergent in mean to ζ.

Theorem 2.

If $\{{\varrho }_j\}$ and $\{{\varsigma }_j\}$ are the real and imaginary parts of the sequence $\{{\zeta }_j\}$, respectively, and both are lacunary almost statistically convergent of order $\alpha$ in measure to ϱ and ς, respectively, then the sequence $\{{\zeta }_j\}$ is also lacunary almost statistically convergent of order $\alpha$ in measure to $\zeta =\varrho +i\varsigma$.

Proof. 

The sequences $\{{\varrho }_j\}$ and $\{{\varsigma }_j\}$ are lacunary almost statistically convergent in measure to $\varrho$ and $\varsigma$. Then, $\forall \epsilon ,\delta >0$, and we get

$${\displaystyle \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel g_j{,}_k-\varrho \parallel \ge \frac{\epsilon }{\sqrt{2}}\right)\ge \delta \left\}\right|=0}$$

and

$${\displaystyle \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel h_j{,}_k-\varsigma \parallel \ge \frac{\epsilon }{\sqrt{2}}\right)\ge \delta \left\}\right|=0,}$$

where ${\displaystyle g_j{,}_k=\begin{array}{c}\hfill \frac{1}{j+1}\end{array}{\displaystyle \sum _{l=0}^j}{\varrho }_{k+l}}$ and ${\displaystyle h_j{,}_k=\frac{1}{j+1}{\displaystyle \sum _{l=0}^j}{\varsigma }_{k+l}}$.

Also, by letting

$$\parallel f_j{,}_k-\zeta \parallel =\sqrt{{(g_j{,}_k-\varrho )}^2+{(h_j{,}_k-\varsigma )}^2},$$

it follows from the contrapositive argument that if this norm is at least $\epsilon$, then at least one of the components $(g_j{,}_k)$ or $(h_j{,}_k)$ must deviate from its limit by at least ${\displaystyle \frac{\epsilon }{\sqrt{2}}}$.

Therefore, we get the inclusion

$$\{\parallel f_j{,}_k-\zeta \parallel \ge \epsilon \}\subset \left\{\right(|g_j{,}_k-\varrho |\ge {\displaystyle \frac{\epsilon }{\sqrt{2}}})\cup (|h_j{,}_k-\varsigma |\ge {\displaystyle \frac{\epsilon }{\sqrt{2}}}\left)\right\}.$$

Using the subadditivity of an uncertain measure, it follows that

$$\begin{array}{cc}& \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel f_j{,}_k-\zeta \parallel \ge \epsilon \right)\ge \delta \left\}\right|\hfill \\ \hfill \le & \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel g_j{,}_k-\varrho \parallel \ge {\displaystyle \frac{\epsilon }{\sqrt{2}}}\right)\ge \delta \left\}\right|\hfill \\ \hfill +& \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel h_j{,}_k-\varsigma \parallel \ge {\displaystyle \frac{\epsilon }{\sqrt{2}}}\right)\ge \delta \left\}\right|=0.\hfill \end{array}$$

This implies that the sequence $({\zeta }_j)$ is lacunary almost statistically convergent in measure to $\zeta$. □

Theorem 3.

If $\{{\varrho }_j\}$ and $({\varsigma }_j)$ are the real and imaginary parts of the sequence $\{{\zeta }_j\}$, respectively, and both are lacunary almost statistically convergent of order $\alpha$ in measure to ϱ and ς, respectively, then the sequence $({\zeta }_j)$ is lacunary almost statistically convergent of order $\alpha$ in distribution to $\zeta =\varrho +i\varsigma$.

Proof. 

Consider $v=e+if$ to be a given point of continuity for a complex uncertain distribution $\varphi$. Then, for any $\beta >e$, $\gamma >f$, we have

$$\begin{array}{ccc}\hfill \{g_j{,}_k\le e,h_j{,}_k\le f\}& =& \{g_j{,}_k\le e,h_j{,}_k\le f,\varrho \le \beta ,\varsigma \le \gamma \}\hfill \\ & & \cup \{g_j{,}_k\le e,h_j{,}_k\le f,\varrho >\beta ,\varsigma >\gamma \}\hfill \\ & & \cup \{g_j{,}_k\le e,h_j{,}_k\le f,\varrho \le \beta ,\varsigma >\gamma \}\hfill \\ & & \cup \{g_j{,}_k\le e,h_j{,}_k\le f,\varrho >\beta ,\varsigma \le \gamma \}\hfill \\ & & \subset \{\varrho \le \beta ,\varsigma \le \gamma \}\cup \{|g_j{,}_k-\varrho |\ge \beta -e\}\hfill \\ & & \cup \{|h_j{,}_k-\varsigma |\ge \gamma -f\}.\hfill \end{array}$$

By the subadditivity axiom, we get

$$\begin{array}{ccc}\hfill {\varphi }_j{,}_k(v)& =& {\varphi }_j{,}_k(e+if)\hfill \\ & \le & \varphi (\beta +i\gamma )+\widehat{\mathit{M}}\{|g_j{,}_k-\varrho |\ge \beta -e\}+\widehat{\mathit{M}}\{|h_j{,}_k-\varsigma |\ge \gamma -f\},\hfill \end{array}$$

since $({\varrho }_j)$ and $({\varsigma }_j)$ are lacunary almost statistically convergent in measure to $\varrho$ and $\varsigma$, respectively. For a given $\epsilon$$>0$ and $j\in I_r$, we have

$${\displaystyle \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel g_j{,}_k-\varrho \parallel \ge \beta -e\right)\ge \epsilon \left\}\right|=0}$$

and

$${\displaystyle \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel h_j{,}_k-\varsigma \parallel \ge \gamma -f\right)\ge \epsilon \left\}\right|=0.}$$

Now, we get

$$\underset{j\to \infty }{\lim }\sup {\varphi }_j{,}_k(v)\le {\varphi }_j{,}_k(\beta +i\gamma ).$$

For any $\beta >e$ and $\gamma >f$, assuming $\beta +i\gamma \to e+if$ and the fact that distribution function $\varphi$ is right continuous, we get

$$\begin{array}{c}\hfill \underset{j\to \infty }{\lim }\sup {\begin{array}{c}\hfill \varphi \end{array}}_j{,}_k(v)\le \varphi (v).\end{array}$$

(1)

On the other hand,

$$\begin{array}{ccc}\hfill \{\varrho \le p,\varsigma \le q\}& =& \{g_j{,}_k\le e,h_j{,}_k\le f,\varrho \le p,\varsigma \le q\}\hfill \\ & & \cup \{g_j{,}_k\le e,h_j{,}_k>f,\varrho \le p,\varsigma \le q\}\hfill \\ & & \cup \{g_j{,}_k>e,h_j{,}_k\le f,\varrho \le p,\varsigma \le q\}\hfill \\ & & \cup \{g_j{,}_k>e,h_j{,}_k>f,\varrho \le p,\varsigma \le q\}\hfill \\ & & \subset \{g_j{,}_k\le e,h_j{,}_k\le f\}\cup \hfill \\ & & \{|g_j{,}_k-\varrho |\ge e-p\}\hfill \\ & & \cup \{|h_j{,}_k-\varsigma |\ge f-q\},\hfill \end{array}$$

which means that

$$\begin{array}{cc}\hfill \varphi (p+iq)& \le {\varphi }_j{,}_k(e+if)+\widehat{\mathit{M}}\{|g_j{,}_k-\varrho |\ge e-p\}\\ & +\widehat{\mathit{M}}\{|h_j{,}_k-\varsigma |\ge f-q\}.\end{array}$$

For a preassigned $\epsilon$$>0$

$${\displaystyle \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel g_j{,}_k-\varrho \parallel \ge e-p\right)\ge \epsilon \left\}\right|=0}$$

and

$${\displaystyle \underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel h_j{,}_k-\varsigma \parallel \ge f-q\right)\ge \epsilon \left\}\right|=0.}$$

So, we have

$$\begin{array}{ccc}\hfill \begin{array}{c}\hfill \varphi (p+iq)\end{array}& \le & \underset{j\to \infty }{\lim \inf } {\varphi }_j{,}_k(e+if)\hfill \end{array}$$

for any $p<e,$$q<f.$ Taking $p+iq\to e+if,$ we get

$$\begin{array}{c}\hfill \varphi (v)\le \underset{j\to \infty }{\lim \inf } {\varphi }_j{,}_k(v).\end{array}$$

(2)

From inequalities (1) and (2), we have that ${\varphi }_j{,}_k(v)$ is lacunary almost statistically convergent in distribution to $\varphi (v)$ as $j\to \infty$. The converse of the above theorem is not true. □

Remark 2.

Lacunary almost statistical convergence of order $\alpha$ in distribution does not imply lacunary almost statistical convergence of order $\alpha$ in measure. An explanation is provided in the following example.

Example 3.

Consider the uncertainty space $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ to be $\{{\omega }_1,{\omega }_2\}$ with $\widehat{\mathit{M}}({\omega }_1)=0.5,$ $\widehat{\mathit{M}}({\omega }_2)=0.5.$ Define the complex uncertain variable as

$$\zeta (\omega )=\left\{\begin{array}{cc}1,\hfill & if\omega ={\omega }_1;\hfill \\ -1,\hfill & if\omega ={\omega }_2;\hfill \\ 0.5\hfill & ifotherwise.\hfill \end{array}\right.$$

Define ${\zeta }_j=-\zeta$ for $j=1,2,\dots .$ Then, ${\zeta }_j$ and ζ have the same distribution as follows:

$${\varphi }_j(v)=\begin{array}{c}\hfill {\varphi }_j(e+if)\end{array}=\left\{\begin{array}{cc}0,& ife<0,-\infty <f<+\infty ;\\ 0,& ife\ge 0,f<-1;\\ 0.5,& ife\ge 0,-1\le f<1;\\ 1,& ife\ge 0,f\ge 1.\end{array}\right.$$

Since each ${\zeta }_j=-\zeta$ the values taken by ${\zeta }_j$ are exactly the same as those taken by $\zeta$, just with opposite signs. However, because the distribution only depends on the probability (or uncertain measure) of these values and not on their signs, the distributions of ${\zeta }_j$ and $\zeta$ are the same. This is why the sequence ${\zeta }_j$ converges in distribution to $\zeta$. So, $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ in distribution to ζ.

For any $\epsilon >0,$

$$\begin{array}{cc}\hfill \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}|\{j\in I_r:& \widehat{\mathit{M}}\parallel f_j{,}_k-\zeta \parallel \ge \epsilon )\}|\hfill \\ \hfill =\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}|\{j\in I_r:& \widehat{\mathit{M}}(\omega \in \mathrm{\Omega }:\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon )\}|\hfill \\ \hfill since\mathrm{\Omega }=\{{\omega }_1,{\omega }_2\}with\widehat{\mathit{M}}({\omega }_1)=0.5,\widehat{\mathit{M}}({\omega }_2)=0.5.\\ \hfill So,\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}|\{j\in I_r:& \widehat{\mathit{M}}(\omega \in \mathrm{\Omega }:\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon )\}|\hfill \\ \hfill & =1.\hfill \end{array}$$

Hence, $\{{\zeta }_j\}$ is not lacunary almost statistically convergent of order $\alpha$ in measure to ζ.

Corollary 1.

If the uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ in measure to $\zeta$, then the uncertain sequence is also lacunary almost statistically convergent of order $\alpha$ in distribution.

Proof. 

By using Theorems 2 and 3, we can easily prove the desired result. □

Corollary 2.

Lacunary almost statistical convergence of order $\alpha$ in mean implies lacunary almost statistical convergence of order $\alpha$ in distribution.

Proof. 

By using Corollary 1 and Theorem 1, we can easily prove the desired result. □

Remark 3.

If the uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ almost surely to ζ, then it does not imply lacunary almost statistical convergence of order $\alpha$ in mean. Now, let us look at an example given below.

Example 4.

Consider the uncertainty space $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ to be $\{{\omega }_1,{\omega }_2,\dots \}$ with ${\displaystyle \widehat{\mathit{M}}(\mathrm{\Lambda })=\sum _{{\omega }_j\in \mathrm{\Lambda }}\frac{1}{2^j}.}$

Define the complex uncertain variables by

$${\zeta }_j(\omega )=\left\{\begin{array}{cc}i{2}^j,\hfill & if\omega ={\omega }_j;forj\in \mathbb{N}\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

for $j\in \mathbb{N}$ and $\zeta \equiv 0.$ Thus, the sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent almost surely to $\zeta .$

Consider

$${\varphi }_j{,}_k(x)=\left\{\begin{array}{cc}0,\hfill & ifx<0;\hfill \\ \\ 1-{\displaystyle \frac{1}{2^j}},\hfill & if0\le x<2^j;\hfill \\ \\ 1,\hfill & ifx\ge 2^j,\hfill \end{array}\right.$$

where ${\varphi }_j{,}_k(x)$ is an uncertainty distribution of $\parallel {\zeta }_j\parallel$.

$$\begin{array}{cc}\hfill \mathit{Now},\underset{r\to \infty }{\lim }\frac{1}{{(j_r-j_{r-1})}^{\alpha }}& \left|\right\{j\in I_r:\mathbf{E}\left(\parallel f_j{,}_k-\zeta \parallel \right)\left\}\right|\hfill \\ \hfill =& {\int }_0^{+\infty }1-\{{\varphi }_j{,}_k(x)\}dx-{\int }_{-\infty }^0\{{\varphi }_j{,}_k(x)\}dx\hfill \\ \hfill =& {\int }_0^{2^j}1-(1-{\displaystyle \frac{1}{2^j}})dx+{\int }_{2^j}^{+\infty }(1-1)dx-{\int }_{\infty }^{0}0dx\hfill \\ \hfill =& {\int }_0^{2^j}{\displaystyle \frac{1}{2^j}}dx\hfill \\ \hfill =& 1.\hfill \end{array}$$

So, the sequence $\{{\zeta }_j\}$ is not lacunary almost statistically convergent of order $\alpha$ in mean to ζ.

Remark 4.

If the complex uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ almost surely to ζ, then it may not be lacunary almost statistically convergent of order $\alpha$ in measure. Here is an example to demonstrate this.

Example 5.

Consider the uncertainty space $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ to be $\{{\omega }_1,{\omega }_2,{\omega }_3\dots \dots \dots \}.$

Define

$$\widehat{\mathit{M}}\{\mathrm{\Lambda }\}=\left\{\begin{array}{cc}{\sup }_{{\omega }_{j\in \mathrm{\Lambda }}}{\displaystyle \frac{j}{(2j+1)}},\hfill & if{\sup }_{{\omega }_{j\in \mathrm{\Lambda }}}{\displaystyle \frac{j}{(2j+1)}}<{\displaystyle \frac{1}{2}};\hfill \\ \\ 1-{\sup }_{{\omega }_{j\in {\mathrm{\Lambda }}^c}}{\displaystyle \frac{j}{(2j+1)}},\hfill & if{\sup }_{{\omega }_{j\in {\mathrm{\Lambda }}^c}}{\displaystyle \frac{j}{(2j+1)}}<{\displaystyle \frac{1}{2}};\hfill \\ \\ 0.5,\hfill & otherwise.\hfill \end{array}\right.$$

Now, we define the complex uncertain variable as

$${\zeta }_j(\omega )=\left\{\begin{array}{cc}ij,\hfill & if\omega ={\omega }_j;\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

for $j=1,2,3,\dots$ and $\zeta \equiv 0.$ Then, the sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ almost surely to ζ. Also, for $\epsilon >0$, we get

$$\begin{array}{ccc}& =& \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}(\parallel f_j{,}_k-\zeta \parallel \ge \epsilon )\left\}\right|\hfill \\ & =& \underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}(\omega \in \mathrm{\Omega }:\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon )\left\}\right|\hfill \\ & =& \underset{r\to \infty }{\lim }\widehat{\mathit{M}}({\omega }_j)\hfill \\ & =& {\displaystyle \frac{j}{2j+1}}⟶{\displaystyle \frac{1}{2}}\hfill \end{array}$$

as $j\to \infty$. Thus, $\{{\zeta }_j\}$ is not lacunary almost statistically convergent of order $\alpha$ in measure to ζ.

Remark 5.

If the uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ in measure to ζ, then it may not be lacunary almost statistically convergent of order $\alpha$ almost surely. Let us consider an example, which is given below.

Example 6.

Consider the uncertainty space $(\mathrm{\Omega },\ell ,\widehat{\mathit{M}})$ to be $[0,1]$ with a Borel $\sigma$- algebra and $\widehat{M}$ to be the Lebesgue measure. For the positive integer $j\in {\mathbb{Z}}^{+}$, there is an integer o such that $j=2^o+N,$ where N is an integer between 0 and $2^o-1$.

The complex uncertain variable is defined as

$${\zeta }_j(\omega )=\left\{\begin{array}{cc}i,\hfill & if{\displaystyle \frac{N}{2^o}}<\omega \le {\displaystyle \frac{N+1}{2^o}};\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

for $j=1,2,3,\dots ,$$N\in \mathbb{Z}$ and $\zeta \equiv 0.$ For some given $\epsilon ,\delta >0$ and $j\ge 2,$ one can have

$$\begin{array}{cc}\hfill \underset{r\to \infty }{\lim }& {\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}(\parallel f_j{,}_k-\zeta \parallel \ge \epsilon )\ge \delta \left\}\right|\hfill \\ \hfill =\underset{r\to \infty }{\lim }& {\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}(\omega :\parallel f_j{,}_k(\omega )-\zeta (\omega )\parallel \ge \epsilon )\ge \delta \left\}\right|\hfill \\ \hfill =\underset{r\to \infty }{\lim }& {\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{\mathit{M}}\left(\parallel {\omega }_j\parallel \right)\ge \delta \left\}\right|=0.\hfill \end{array}$$

Thus, $\{{\zeta }_j\}$ is lacunary almost statistically convergent in measure to ζ. Further, we have for $\epsilon \ge 0$

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\mathbf{E}\left(\parallel f_j{,}_k-\zeta \parallel \right)\ge \epsilon \left\}\right|=0.$$

Therefore, sequence $\{{\zeta }_j\}$ is also lacunary almost statistically convergent in mean to ζ. As, for any $\omega \in [0,1],$ there exists an infinite number of closed intervals that are of the form $[{\displaystyle \frac{N}{2^o}},{\displaystyle \frac{N+1}{2^o}}]$ containing $\omega \in \mathrm{\Lambda }$, where $\mathrm{\Lambda }$ represents an uncertain event, $\widehat{\mathit{M}}$($\mathrm{\Lambda }$) = 1.

Thus, $\{{\zeta }_j\}$ is not lacunary almost statistically convergent of order $\alpha$ almost surely to $\zeta .$

Proposition 1.

The complex uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ almost surely and uniformly in k to $\zeta$ if and only if for every $\epsilon >0$ and $\delta >0$,

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{n=1}^{\infty }}{\displaystyle \bigcap _{m=n}^{\infty }}\right\{\omega \in \mathrm{\Omega }:\underset{k}{\sup }\parallel f_{j+m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right|\ge \delta \left)\right\}=0.$$

Proof. 

Suppose that the sequence $\{f_{j,k}(\omega )\}$ converges to $\zeta (\omega )$ almost surely and uniformly in k in the sense of lacunary almost statistical convergence of order $\alpha$. Then, there exists a set $\mathrm{\Lambda }\subseteq \mathrm{\Omega }$ with $\widehat{M}(\mathrm{\Lambda })=1$ such that for every $\omega \in \mathrm{\Lambda }$ and every $\epsilon >0$, there exists $N=N(\omega ,\epsilon )\in \mathbb{N}$ such that

$$\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel <\epsilon ,\mathrm{for}\mathrm{all}j\ge N.$$

This implies

$$\omega \in {\displaystyle \bigcap _{n=1}^{\infty }}{\displaystyle \bigcup _{j=n}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel <\epsilon \}.$$

Define the set

$$A:={\displaystyle \bigcap _{n=1}^{\infty }}{\displaystyle \bigcup _{j=n}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel <\epsilon \}.$$

Then, clearly, $\widehat{M}(A)=1$. By the duality axiom of an uncertain measure, we have

$$\widehat{M}(A^c)=\widehat{M}\left({\displaystyle \bigcup _{n=1}^{\infty }}{\displaystyle \bigcap _{j=n}^{\infty }}\right\{\omega :\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right)=0.$$

Now, fix any $j\in I_r$ and define the event

$$A_j:={\displaystyle \bigcup _{n=1}^{\infty }}{\displaystyle \bigcap _{m=n}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{j+m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \}.$$

Then, clearly, $\widehat{M}(A_j)=0$ for all $j\in \mathbb{N}$. Therefore, the number of indices $j\in I_r$ for which $\widehat{M}(A_j)\ge \delta$ becomes negligible in lacunary density of order $\alpha$, that is,

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}(A_j)\ge \delta \left\}\right|=0.$$

This gives the desired result. Conversely, suppose that for every $\epsilon >0$ and $\delta >0$, we have

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{n=1}^{\infty }}{\displaystyle \bigcap _{m=n}^{\infty }}\right\{\omega \in \mathrm{\Omega }:\underset{k}{\sup }\parallel f_{j+m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right|\ge \delta \left)\right\}=0.$$

Then, for each fixed $j\in \mathbb{N}$, define

$$A_j:={\displaystyle \bigcup _{n=1}^{\infty }}{\displaystyle \bigcap _{m=n}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{j+m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \},$$

and we have $\widehat{M}(A_j)=0$ for “almost all” j in lacunary density of order $\alpha$. Define the set

$$A:=\{\omega \in \mathrm{\Omega }:\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel ⇸0\mathrm{as}j\to \infty \}.$$

Then, for each $\omega \in A$, there exists $\epsilon >0$ such that for infinitely many j,

$$\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon ,$$

so $\omega$ must belong to infinitely many $A_j$. But since the set of $j\in I_r$ with $\widehat{M}(A_j)\ge \delta$ has lacunary density of order $\alpha$ equal to zero, it follows that $\widehat{M}(A)=0$. Hence, there exists a subset $\mathrm{\Lambda }=\mathrm{\Omega }\setminus A$ with $\widehat{M}(\mathrm{\Lambda })=1$ such that for each $\omega \in \mathrm{\Lambda }$,

$$\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \to 0,$$

that is, $\{f_{j,k}(\omega )\}$ converges to $\zeta (\omega )$ uniformly in k and almost surely in the sense of lacunary statistical convergence of order $\alpha$. □

Proposition 2.

The complex uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ uniformly almost surely to $\zeta$ if and only if for every $\epsilon >0$ and $\delta >0$, we have

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega \in \mathrm{\Omega }:\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right|\ge \delta \left\}\right)=0.$$

Proof. 

Suppose that $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ uniformly almost surely to $\zeta$. Then, there exists a set $J\subseteq \mathrm{\Omega }$ such that $\widehat{M}(J)<\kappa$ for arbitrary $\kappa >0$, and on $\mathrm{\Omega }\setminus J$, we have uniform convergence in k, i.e., for every $\epsilon >0$, there exists $n=n(\omega ,\epsilon )$ such that

$$\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel <\epsilon \mathrm{for}\mathrm{all}m\ge n\mathrm{and}\omega \in \mathrm{\Omega }\setminus J.$$

This implies that for such $\omega$ and any $j\ge n$,

$$\omega \notin {\displaystyle \bigcup _{m=j}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \},$$

and hence

$$\widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right)<\kappa <\delta .$$

So, for sufficiently large r, the number of such $j\in I_r$ with this measure exceeding $\delta$ becomes negligible:

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right|\ge \delta \left\}\right)=0.$$

Conversely, suppose the above condition holds. For each $\ell \in \mathbb{N}$, let $\epsilon =1/\ell$ and define ${\delta }_{\ell }={\displaystyle \frac{\delta }{2^{\ell }}}$. Then, for each ℓ, the condition gives

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge {\displaystyle \frac{1}{\ell }}\left\}\right|\ge {\delta }_{\ell }\left\}\right)=0.$$

This implies that for each ℓ, there exists a set $J_{\ell }\subseteq \mathrm{\Omega }$ such that

$$\widehat{M}(J_{\ell })<{\delta }_{\ell },$$

and for all $\omega \in \mathrm{\Omega }\setminus J_{\ell }$, there exists $n_{\ell }=n_{\ell }(\omega )$ such that

$$\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel <{\displaystyle \frac{1}{\ell }}\mathrm{for}\mathrm{all}m\ge n_{\ell }.$$

Define the exceptional set

$$J={\displaystyle \bigcup _{\ell =1}^{\infty }}{J}_{\ell },$$

Then, by subadditivity,

$$\widehat{M}(J)\le {\displaystyle \sum _{\ell =1}^{\infty }}\widehat{M}(J_{\ell })<{\displaystyle \sum _{\ell =1}^{\infty }}{\displaystyle \frac{\delta }{2^{\ell }}}=\delta .$$

On $\mathrm{\Omega }\setminus J$, the sequence $\{f_{m,k}(\omega )\}$ converges uniformly in k to $\zeta (\omega )$, i.e., the convergence is uniform almost surely. Therefore, $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ uniformly almost surely to $\zeta$. □

Theorem 4.

If the complex uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ uniformly almost surely to $\zeta$, then the sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ almost surely to $\zeta$.

Proof. 

By Proposition 2, for every $\epsilon >0$ and $\delta >0$, we have

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega \in \mathrm{\Omega }:\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right|\ge \delta \left\}\right)=0.$$

Now, fix any $k\in \mathbb{N}$. Observe that for each $m\ge j$,

$$\{\omega :\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \}\subseteq \{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \}.$$

Taking the union over $m\ge j$ and applying the uncertainty measure, we obtain

$$\widehat{M}({\displaystyle \bigcup _{m=j}^{\infty }}\{\omega :\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \})\le \widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right).$$

Therefore, the set

$$\{j\in I_r:\widehat{M}({\displaystyle \bigcup _{m=j}^{\infty }}\left\{\omega :\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \right\})\ge \delta \}$$

is contained in

$$\{j\in I_r:\widehat{M}({\displaystyle \bigcup _{m=j}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \})\ge \delta \}.$$

Since the lacunary density of order $\alpha$ of the right-hand set tends to zero as $r\to \infty$, the same holds for the left-hand set. Hence, $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ almost surely to $\zeta$. □

Theorem 5.

If the complex uncertain sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ uniformly almost surely to $\zeta$, then the sequence $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ in measure to $\zeta$.

Proof. 

By Proposition 2, for every $\epsilon >0$ and $\delta >0$, we have

$$\underset{r\to \infty }{\lim }{\displaystyle \frac{1}{{(j_r-j_{r-1})}^{\alpha }}}\left|\right\{j\in I_r:\widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega \in \mathrm{\Omega }:\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right)\ge \delta \left\}\right|=0.$$

Now, observe that for any fixed k and $j\in I_r$, we have

$$\{\omega :\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \}\subseteq \{\omega :\underset{k}{\sup }\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \}\subseteq {\displaystyle \bigcup _{m=j}^{\infty }}\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \}.$$

Taking the uncertainty measure on both sides yields

$$\widehat{M}(\{\omega :\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \})\le \widehat{M}\left({\displaystyle \bigcup _{m=j}^{\infty }}\right\{\omega :\underset{k}{\sup }\parallel f_{m,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \left\}\right).$$

Thus, the index set

$$\{j\in I_r:\widehat{M}(\{\omega :\parallel f_{j,k}(\omega )-\zeta (\omega )\parallel \ge \epsilon \})\ge \delta \}$$

is a subset of the index set in Proposition 2, which has lacunary density of order $\alpha$ tending to zero. Hence, $\{{\zeta }_j\}$ is lacunary almost statistically convergent of order $\alpha$ in measure to $\zeta$. □

4. Conclusions

Here, we have studied the concept of lacunary almost statistical convergence in measure, convergence in mean, convergence in distribution, convergence almost surely and convergence uniformly almost surely of complex uncertain sequences. We also illustrated the interconnections between these convergence types using graphical representations, providing a clearer understanding of how uncertain sequences behave under various convergence frameworks. In many real-world situations where exact data is unavailable or imprecise, classical methods of function approximation may fail. The convergence concepts explored in this study offer a more robust and flexible approach to approximating uncertain or complex functions. By extending the tools of approximation theory into the setting of uncertainty, our findings provide a foundation for developing more reliable and realistic approximation techniques in uncertain environments. One can use these ideas in other areas of study and in a broader sense.