Asymptotically λ-invariant statistical equivalent sequences of fuzzy numbers

Abstract

This paper presents the following definition which is a natural combination of the definitions for asymptotically equivalent λ-statistical convergence and σ-convergence of fuzzy numbers. Two sequences X and Y of fuzzy numbers are said to be asymptotically λ-invariant statistical equivalent of multiple L provided that for every ε > 0,

$$\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|=0,\text{uniformly in}m$$

$$\left(\text{denoted by}X\stackrel{S_{\sigma ,\lambda }^L(F)}{\sim }Y\right)$$

and simply asymptotically λ-invariant statistical equivalent if$L=\overline{1}$.

Introduction

Let σ be a one-to-one mapping of the set of positive integers into itself such that σ^k(m) = σ (σ^k−1(m)), k = 1,2,3,…. The generalized de la Vallee-Pousin mean is defined by

$$t_n\left(x\right)=\frac{1}{{\lambda }_n}\sum _{k\in I_n}{x}_k,$$

where (λ _n ) is a non-decreasing sequence of positive numbers such that λ_{n + 1}≤ λ _n + 1,λ₁ = 1, and λ _n → ∞ as n → ∞ and I _n = n− λ _n + 1,n. A sequence x = (x _k ) is said to be (V,λ)-summable to a number L if t _n (x) → L as n → ∞[1]. (V λ)-summability reduces to (C,1)-summability when λ _n = n for all n ∈ N.

Let D denote the set of all closed and bounded intervals on$\mathbb{R}$, the real line. For X,Y ∈ D, we define

$$d\left(X,Y\right)=max\left(\left|a_1-b_1\right|,\left|a_2-b_2\right|\right),$$

where X = [a₁,a₂] and Y = [b₁,b₂]. It is known that (D,d) is a complete metric space. A fuzzy real number X is a fuzzy set on$\mathbb{R}$, i.e., a mapping$X:\mathbb{R}\to I\left(=\left[0,1\right]\right)$ associating each real number t with its grade of membership X(t).

The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted by$\mathbb{R}\left(I\right)$. Throughout the paper, by a fuzzy real number X, we mean that$X\in \mathbb{R}\left(I\right)$.

The α-cut or α-level set [X]^α of the fuzzy real number X, for 0 < α ≤ 1, is defined by${\left[X\right]}^{\alpha }=\left\{t\in \mathbb{R}:X\left(t\right)\ge \alpha \right\};$ for α = 0, it is the closure of the strong 0-cut, i.e., closure of the set {t ∈ R : X (t} > 0). The linear structure of$\mathbb{R}\left(I\right)$ induces the addition X + Y and the scalar multiplication μX,$\mu \in \mathbb{R}$, in terms of α-level sets, defined by

$${\left[X+Y\right]}^{\alpha }={\left[X\right]}^{\alpha }+{\left[Y\right]}^{\alpha },{\left[\mu X\right]}^{\alpha }=\mu {\left[X\right]}^{\alpha }$$

for each α ∈ (0,1].

Let$\stackrel{-}{d}:\mathbb{R}\left(I\right)\times \mathbb{R}\left(I\right)\to \mathbb{R}$ be defined by

$$\stackrel{-}{d}\left(X,Y\right)=\underset{0\le \alpha \le 1}{sup}d\left({\left[X\right]}^{\alpha },{\left[Y\right]}^{\alpha }\right).$$

Then,$\stackrel{-}{d}$ defines a metric on$\mathbb{R}\left(I\right)$. It is well known that$\mathbb{R}\left(I\right)$ is complete with respect to$\stackrel{-}{d}$.

A sequence (X _k ) of fuzzy real numbers is said to be convergent to the fuzzy real number X₀ if, for every ε > 0, there exists n₀ ∈ N such that$\stackrel{-}{d}\left(X_k,X_0\right)<\epsilon$, for all k ≥ n₀. Let c(F) denote the set of all convergent sequences of fuzzy numbers.

A sequence (X _k ) of fuzzy real numbers is said to be bounded if the set {X _k : k ∈ N} of fuzzy numbers is bounded. We denote by ℓ _∞ (F) the set of all bounded sequences of fuzzy numbers. In[2], it was shown that c(F) and ℓ _∞ (F) are complete metric spaces.

A subset E of N is said to have density (asymptotic or natural) δ(E) if

$$\delta \left(E\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{n}\sum _{k=1}^n{\varkappa }_E\left(k\right)\text{exists,}$$

where ϰ _E is the characteristic function of E. The definition of statistical convergence was introduced by Fast[2] and studied by several authors[3–9]. The sequence x is statistically convergent to s if for each ε > 0,

$$\underset{n\to \infty }{\text{lim}}\frac{1}{n}\left|\{k\le n:|x_k-s|\ge \epsilon \}\right|=0,$$

where |A| denotes the number of elements in A. Schoenberg[10] studied statistical convergence as a summability method and listed some of the elementary properties of statistical convergence.

Nuray and Savaş[11] defined the notion of statistical convergence for sequences of fuzzy real numbers and studied some properties. A fuzzy real number (X _k ) is said to be statistically convergent to the fuzzy real number X₀ if for every ε > 0,

$$\delta \left(\left\{k\in N:\stackrel{-}{d}\left(X_k,X_0\right)\ge \epsilon \right\}\right)=0.$$

Fuzzy sequence are spaces studied by several authors such as[12–19].

In 1993, Marouf[20] presented definitions for asymptotically equivalent sequences of real numbers and asymptotic regular matrices. In 2003, Patterson[21] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. In 2006, Savaş and Başarir[22] introduced and studied the concept of (σ λ)-asymptotically statistical equivalent sequences. In 2008, Esi and Esi[23] introduced and studied the concept of asymptotically equivalent difference sequences of fuzzy numbers. In 2009, Esi[24] introduced and studied asymptotically equivalent sequences for double sequences. For sequences of fuzzy numbers, Savaş[25, 26] introduced and studied the concepts of strongly λ-summable λ-statistical convergence and asymptotically λ-statistical equivalent sequences, respectively . Recently, Braha[27] defined asymptotically generalized difference lacunary sequences. The goal of this paper is to extend the idea on asymptotically equivalent and${\lambda }_{{\sigma }^F}$-statistical convergence of fuzzy numbers.

Methods

Definitions and notations

Definition 1

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be σ^F-asymptotically equivalent if

$$\begin{array}{l}\underset{k}{\text{lim}}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},\overline{1}\right)\\ =0\text{, uniformly in}m\left(\text{denoted by}X\stackrel{{\sigma }^F}{\sim }Y\right).\end{array}$$

Definition 2

A sequence of fuzzy numbers, X = (X _k ), is said to be$S_{\sigma ,\lambda }^L\left(F\right)$-statistically convergent or$S_{{\sigma }^F}^{\lambda }$-convergent to the fuzzy number L if for every ε > 0,

$$\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(X_{{\sigma }^k\left(m\right)},L\right)\ge \epsilon \right\}\right|=0,\text{uniformly in}m.$$

In this case, we write$S_{{\sigma }^F}^{\lambda }-limX=L$ or$X_k\to L\left(S_{{\sigma }^F}^{\lambda }\right)$.

Following this result, we introduce two new notions asymptotically$S_{\sigma ,\lambda }^L\left(F\right)$-statistical equivalent of multiple L and strong$V_{\sigma ,\lambda }^L\left(F\right)$-asymptotically equivalent of multiple L.

The next definition is a natural combination of Definitions 1 and 2.

Definition 3

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be asymptotically λ-invariant statistical equivalent of multiple L provided that for every ε > 0,

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|\\ =0\text{, uniformly in}m\left(\text{denoted by}X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y\right)\end{array}$$

and simply asymptotically$S_{\sigma ,\lambda }^{}\left(F\right)$-statistical equivalent if$L=\overline{1}$.

Example 1

Let λ _n = n and σ(m) = m + 1 for all$n,m\in \mathbb{N}$. Consider the sequences of fuzzy numbers X = (X _k ) and Y = (Y _k ) defined by$X_n=\overline{n^{-2}}$ and$Y_n=\overline{n^{-1}}$ for all$n\in \mathbb{N}$. Then,

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|\\ =\underset{n}{\text{lim}}\frac{1}{n}\left|\left\{k\in \left[1,n\right]:\stackrel{-}{d}\left(\overline{n^{-1}},\overline{0}\right)\ge \epsilon \right\}\right|=0.\end{array}$$

If we take λ _n = n for all$n\in \mathbb{N}$, the above definition reduces to following definition:

Definition 4

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be asymptotically invariant statistical equivalent of multiple L provided that for every ε > 0,

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{n}\left|\left\{k\le n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|\\ =0\text{, uniformly in}m\left(\text{denoted by}X\stackrel{S_{\sigma }^L\left(F\right)}{\sim }Y\right)\end{array}$$

and simply asymptotically S _σ (F)-statistical equivalent if$L=\overline{1}$.

Definition 5

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be strong$V_{\sigma ,\lambda }^L\left(F\right)$-asymptotically equivalent of multiple L provided that

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\sum _{k\in I_n}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\\ =0,\text{uniformly in}m\left(\text{denoted by}X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y\right)\end{array}$$

and simply strong$V_{\sigma ,\lambda }^{}\left(F\right)$-asymptotically statistical equivalent if$L=\overline{1}$.

Example 2

Let λ _n = n and σ(m) = m + 1 for all$n,m\in \mathbb{N}$. Consider the sequences of fuzzy numbers X = (X _k ) and Y = (Y _k ) defined by$X_n=\overline{n^{-2}}$ and$Y_n=\overline{n^{-1}}$ for all$n\in \mathbb{N}$. Then,

$$\begin{array}{ll}\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\sum _{k\in I_n}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},\overline{0}\right)& =\underset{n}{\text{lim}}\frac{1}{n}\sum _{k=1}^n\stackrel{-}{d}\left(\frac{\overline{n^{-2k}}}{\overline{n^{-k}}},\overline{0}\right)\\ =\underset{n}{\text{lim}}\frac{1}{n}\sum _{k=1}^n\frac{1}{n^k}<\infty .\end{array}$$

If we take λ _n = n for all$n\in \mathbb{N}$, the above definition reduces to the following definition:

Definition 6

Two sequences X and Y of fuzzy numbers are said to be strong Cesaro$C_{\sigma }^L\left(F\right)$-asymptotically equivalent of multiple L provided that

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{n}\sum _{k=1}^n\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\\ =0,\text{uniformly in}m\left(\text{denoted by}X\stackrel{C_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y\right)\end{array}$$

and simply strong Cesaro C _σ (F)-asymptotically equivalent if$L=\overline{1}$.

If we take σ(m) = m + 1, the above definitions reduce the following definitions:

Definition 7

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be asymptotically almost equivalent if

$$\begin{array}{l}\underset{k}{\text{lim}}\stackrel{-}{d}\left(\frac{X_{k+m}}{Y_{k+m}},\overline{1}\right)\\ =0\text{, uniformly in}m\left(\text{denoted by}X\stackrel{\hat{F}}{\sim }Y\right).\end{array}$$

Definition 8

A sequence of fuzzy numbers X = (X _k ) is said to be${\lambda }_{\hat{F}}$-statistically almost convergent or$S_{\hat{F}}^{\lambda }$-convergent to the fuzzy number L if for every ε > 0,

$$\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(X_{k+m},L\right)\ge \epsilon \right\}\right|=0\text{uniformly in}m.$$

In this case, we write$S_{\hat{F}}^{\lambda }-limX=L$ or$X_k\to L\left(S_{\hat{F}}^{\lambda }\right)$.

Definition 9

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be asymptotically almost λ-statistical equivalent of multiple L provided that for every ε > 0,

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{k+m}}{Y_{k+m}},L\right)\ge \epsilon \right\}\right|\\ =0\text{, uniformly in}m\left(\text{denoted by}X\stackrel{S_{\lambda }^L\left(\hat{F}\right)}{\sim }Y\right)\end{array}$$

and simply asymptotically almost λ-statistical equivalent if$L=\overline{1}$.

If we take λ _n = n for all$n\in \mathbb{N}$, the above definition reduces to the following definition:

Definition 10

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be asymptotically almost statistical equivalent of multiple L provided that for every ε > 0,

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{n}\left|\left\{k\le n:\stackrel{-}{d}\left(\frac{X_{k+m}}{Y_{k+m}},L\right)\ge \epsilon \right\}\right|\\ =0\text{, uniformly in}m\left(\text{denoted by}X\stackrel{S_{}^L\left(\hat{F}\right)}{\sim }Y\right)\end{array}$$

and simply asymptotically almost statistical equivalent if$L=\overline{1}$.

Definition 11

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be strong asymptotically almost λ-equivalent of multiple L provided that

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{{\lambda }_n}\sum _{k\in I_n}\stackrel{-}{d}\left(\frac{X_{k+m}}{Y_{k+m}},L\right)\\ =0,\text{uniformly in}m\left(\text{denoted by}X\stackrel{V_{\lambda }^L\left(\hat{F}\right)}{\sim }Y\right)\end{array}$$

and simply strong asymptotically almost λ-equivalent if$L=\overline{1}$.

If we take λ _n = n for all$n\in \mathbb{N}$, the above definition reduces to following definition.

Definition 12

Two sequences X = (X _k ) and Y = (Y _k ) of fuzzy numbers are said to be strong asymptotically almost equivalent of multiple L provided that

$$\begin{array}{l}\underset{n}{\text{lim}}\frac{1}{n}\sum _{k=1}^n\stackrel{-}{d}\left(\frac{X_{k+m}}{Y_{k+m}},L\right)\\ =0,\text{uniformly in}m\left(\text{denoted by}X\stackrel{C^L\left(\hat{F}\right)}{\sim }Y\right)\end{array}$$

and simply strong asymptotically almost equivalent if$L=\overline{1}$.

Results and discussion

Theorem 1

Let X = (X _k ) and Y = (Y _k ) be two fuzzy real valued sequences. Then, the following conditions are satisfied:

(i) If$X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y,$ then$X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$.

(ii) If X ∈ ℓ _∞ (F) and$X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$, then$X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$; hence,$X\stackrel{C_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$.

(iii) $X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y\cap {\ell }_{\infty }\left(F\right)=X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y\cap {\ell }_{\infty }\left(F\right).$

Proof

(i) If ε > 0 and$X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$, then

$$\sum _{k\in I_n}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \sum _{\genfrac{}{}{0.0pt}{}{k\in I_n}{\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon }}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)$$

$$\ge \epsilon \left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|.$$

Therefore,$X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$.

• (ii) Suppose that X = (X _k ) and Y = (Y _k ) are in ℓ _∞ (F) and $X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$. Then, we can assume that

  $$\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\le T\text{, for all}k\text{and}m.$$

Given ε > 0,

$$\frac{1}{{\lambda }_n}\sum _{k\in I_n}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)=\frac{1}{{\lambda }_n}\sum _{\genfrac{}{}{0.0pt}{}{k\in I_n}{\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon }}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)$$

$$+\frac{1}{{\lambda }_n}\sum _{\genfrac{}{}{0.0pt}{}{k\in I_n}{\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)<\epsilon }}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)$$

$$\le \frac{T}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|+\epsilon .$$

Therefore,$X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$ . Further, we have

$$\begin{array}{l}\frac{1}{n}\sum _{k=1}^n\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\\ =\frac{1}{n}\sum _{k=1}^{n-{\lambda }_n}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)+\frac{1}{n}\sum _{k\in I_n}^{}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\end{array}$$

$$\le \frac{1}{{\lambda }_n}\sum _{k=1}^{n-{\lambda }_n}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)+\frac{1}{{\lambda }_n}\sum _{k\in I_n}^{}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)$$

$$\le \frac{2}{{\lambda }_n}\sum _{k\in I_n}^{}\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right).$$

Hence,$X\stackrel{C_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$ since$X\stackrel{V_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$.

(iii) Follows from (i) and (ii).

□

In the next theorem, we prove the following relation:

Theorem 2

$$X\stackrel{S_{\sigma }^L\left(F\right)}{\sim }Y$$

implies$X\stackrel{S_{\sigma ,\lambda }^L\left(F\right)}{\sim }Y$ if

$$liminf\left(\frac{{\lambda }_n}{n}\right)>0.$$

(1)

Proof

For a given ε > 0, we have

$$\begin{array}{l}\left\{k\le n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\\ \supset \left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}.\end{array}$$

Therefore,

$$\begin{array}{l}\frac{1}{n}\left|\left\{k\le n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|\\ \ge \frac{1}{n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|\end{array}$$

$$\ge \frac{{\lambda }_n}{n}.\frac{1}{{\lambda }_n}\left|\left\{k\in I_n:\stackrel{-}{d}\left(\frac{X_{{\sigma }^k\left(m\right)}}{Y_{{\sigma }^k\left(m\right)}},L\right)\ge \epsilon \right\}\right|.$$

Taking the limit as n → ∞ and using Equation 1, we get the desired result. This completes the proof. □

Conclusions

The concept of asymptotic equivalence was first suggested by Marouf[20] in 1993. After that, several authors introduced and studied some asymptotically equivalent sequences. The results obtained in this study are much more general than those obtained earlier.