On ℐ-asymptotically lacunary statistical equivalent sequences

Abstract

This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐ-statistically limit and ℐ-lacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences $x=(x_k)$ and $y=(y_k)$ are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every $ϵ>0$, and $\delta >0$,

$$\{r\in \mathbb{N}:\frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge \epsilon \}\right|\ge \delta \}\in \mathcal{I}$$

(denoted by $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$) and simply ℐ-asymptotically lacunary statistical equivalent if $L=1$.

MSC:40A99, 40A05.

1 Introduction

In 1993, Marouf [1] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson [2] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.

In [3], asymptotically lacunary statistical equivalent, which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences. Later on, the extension asymptotically lacunary statistical equivalent sequences is presented (see [4]).

Recently, Das, Savaş and Ghosal [5] introduced new notions, namely ℐ-statistical convergence and ℐ-lacunary statistical convergence by using ideal.

In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts ℐ-asymptotically statistical equivalent and ℐ-asymptotically lacunary statistical equivalent. In addition to these definitions, natural inclusion theorems shall also be presented.

First, we introduce some definitions.

2 Definitions and notations

Definition 2.1 (Marouf [1])

Two nonnegative sequences $x=(x_k)$ and $y=(y_k)$ are said to be asymptotically equivalent if

$$\underset{k}{lim}\frac{x_k}{y_k}=1$$

(denoted by $x\sim y$).

Definition 2.2 (Fridy [6])

The sequence $x=(x_k)$ has statistic limit L, denoted by st-$lims=L$ provided that for every $ϵ>0$,

$$\underset{n}{lim}\frac{1}{n}\{\text{the number of }k\le n:|x_k-L|\ge ϵ\}=0.$$

The next definition is natural combination of Definitions 2.1 and 2.2.

Definition 2.3 (Patterson [2])

Two nonnegative sequences $x=(x_k)$ and $y=(y_k)$ are said to be asymptotically statistical equivalent of multiple L provided that for every $ϵ>0$,

$$\underset{n}{lim}\frac{1}{n}\{\text{the number of }k<n:|\frac{x_k}{y_k}-L|\ge ϵ\}=0$$

(denoted by $x\stackrel{S_L}{\sim }y$) and simply asymptotically statistical equivalent if $L=1$.

By a lacunary $\theta =(k_r)$; $r=0,1,2,\dots$ , where $k_0=0$, we shall mean an increasing sequence of nonnegative integers with $k_r-k_{r-1}$ as $r\to \mathrm{\infty }$. The intervals determined by θ will be denoted by $I_r=(k_{r-1},k_r]$ and $h_r=k_r-k_{r-1}$. The ratio $\frac{k_r}{k_{r-1}}$ will be denoted by $q_r$.

Definition 2.4 ([3])

Let θ be a lacunary sequence; the two nonnegative sequences $x=(x_k)$ and $y=(y_k)$ are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every $ϵ>0$

$$\underset{r}{lim}\frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|=0$$

(denoted by $x\stackrel{S_{\theta }^L}{\sim }y$) and simply asymptotically lacunary statistical equivalent if $L=1$.

More investigations in this direction and more applications of asymptotically statistical equivalent can be found in [7, 8] where many important references can be found.

The following definitions and notions will be needed.

Definition 2.5 ([9])

A nonempty family $\mathcal{I}\subset 2^Y$ of subsets a nonempty set Y is said to be an ideal in Y if the following conditions hold:

• (i) $A,B\in \mathcal{I}$ implies $A\cup B\in \mathcal{I}$;

• (ii) $A\in \mathcal{I}$, $B\subset A$ imply $B\in \mathcal{I}$.

Definition 2.6 ([10])

A nonempty family $\mathcal{F}\subset 2^{\mathbb{N}}$ is said to be a filter of ℕ if the following conditions hold:

• (i) $\mathrm{\varnothing }\notin \mathcal{F}$;

• (ii) $A,B\in \mathcal{F}$ implies $A\cap B\in \mathcal{F}$;

• (iii) $A\in \mathcal{F}$, $B\subset A$ imply $B\in \mathcal{F}$.

If ℐ is proper ideal of ℕ (i.e., $\mathbb{N}\notin \mathcal{I}$), then the family of sets $\mathcal{F}(\mathcal{I})=\{M\subset \mathbb{N}:\mathrm{\exists }A\in \mathcal{I}:M=\mathbb{N}\mathrm{\setminus }A\}$ is a filter of ℕ. It is called the filter associated with the ideal.

Definition 2.7 ([9, 10])

A proper ideal ℐ is said to be admissible if $\{n\}\in \mathcal{I}$ for each $n\in \mathbb{N}$.

Throughout ℐ will stand for a proper admissible ideal of ℕ, and by sequence we always mean sequences of real numbers.

Definition 2.8 ([9])

Let $\mathcal{I}\subset 2^{\mathbb{N}}$ be a proper admissible ideal in ℕ.

The sequence $(x_k)$ of elements of ℝ is said to be ℐ-convergent to $L\in \mathbb{R}$ if for each $ϵ>0$ the set $A(ϵ)=\{n\in \mathbb{N}:|x_n-L|\ge ϵ\}\in \mathcal{I}$.

Following these results, we introduce two new notions ℐ-asymptotically lacunary statistical equivalent of multiple L and strong ℐ-asymptotically lacunary equivalent of multiple L.

The following definitions are given in [5].

Definition 2.9 A sequence $x=(x_k)$ is said to be ℐ-statistically convergent to L or $S(\mathcal{I})$-convergent to L if, for any $\epsilon >0$ and $\delta >0$,

$$\{n\in \mathbb{N}:\frac{1}{n}\left|\{k\le n:|x_k-L|\ge \epsilon \}\right|\ge \delta \}\in \mathcal{I}.$$

In this case, we write $x_k\to L(S(\mathcal{I}))$. The class of all ℐ-statistically convergent sequences will be denoted by $S(\mathcal{I})$.

Definition 2.10 Let θ be a lacunary sequence. A sequence $x=(x_k)$ is said to be ℐ-lacunary statistically convergent to L or $S_{\theta }(\mathcal{I})$-convergent to L if, for any $\epsilon >0$ and $\delta >0$,

$$\{r\in \mathbb{N}:\frac{1}{h_r}\left|\{k\in I_r:|x_k-L|\ge \epsilon \}\right|\ge \delta \}\in \mathcal{I}.$$

In this case, we write $x_k\to L(S_{\theta }(\mathcal{I}))$. The class of all ℐ-lacunary statistically convergent sequences will be denoted by $S_{\theta }(\mathcal{I})$.

Definition 2.11 Let θ be a lacunary sequence. A sequence $x=(x_k)$ is said to be strong ℐ-lacunary convergent to L or $N_{\theta }(\mathcal{I})$-convergent to L if, for any $\epsilon >0$

$$\{r\in N:\frac{1}{h_r}\sum _{k\in I_r}|x_k-L|\ge \epsilon \}\in \mathcal{I}.$$

In this case, we write $x_k\to L(N_{\theta }(\mathcal{I}))$. The class of all strong ℐ-lacunary statistically convergent sequences will be denoted by $N_{\theta }(\mathcal{I})$.

3 New definitions

The next definitions are combination of Definitions 2.1, 2.9, 2.10 and 2.11.

Definition 3.1 Two nonnegative sequences $x=(x_k)$ and $y=(y_k)$ are said to be ℐ-asymptotically statistical equivalent of multiple L provided that for every $ϵ>0$ and $\delta >0$,

$$\{n\in N:\frac{1}{n}\left|\{k\le n:|\frac{x_k}{y_k}-L|\ge \epsilon \}\right|\ge \delta \}\in \mathcal{I}$$

(denoted by $x\stackrel{S^L(\mathcal{I})}{\sim }y$) and simply ℐ-asymptotically statistical equivalent if $L=1$.

For $\mathcal{I}={\mathcal{I}}_{\mathrm{fin}}$, ℐ-asymptotically statistical equivalent of multiple L coincides with asymptotically statistical equivalent of multiple L, which is defined in [3].

Definition 3.2 Let θ be a lacunary sequence; the two nonnegative sequences $x=(x_k)$ and $y=(y_k)$ are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every $ϵ>0$ and $\delta >0$,

$$\{r\in \mathbb{N}:\frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge \epsilon \}\right|\ge \delta \}\in \mathcal{I}$$

(denoted by $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$) and simply ℐ-asymptotically lacunary statistical equivalent if $L=1$.

For $\mathcal{I}={\mathcal{I}}_{\mathrm{fin}}$, ℐ-asymptotically lacunary statistical equivalent of multiple L coincides with asymptotically lacunary statistical equivalent of multiple L, which is defined in [3].

Definition 3.3 Let θ be a lacunary sequence; two number sequences $x=(x_k)$ and $y=(y_k)$ are strong ℐ-asymptotically lacunary equivalent of multiple L provided that

$$\{r\in \mathbb{N}:\frac{1}{h_r}\sum _{k\in I_r}|\frac{x_k}{y_k}-L|\ge \epsilon \}\in \mathcal{I}$$

(denoted by $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$) and strong simply ℐ-asymptotically lacunary equivalent if $L=1$.

4 Main result

In this section, we state and prove the results of this article.

Theorem 4.1 Let $\theta =\{k_r\}$ be a lacunary sequence then

• (1)

  • (a) If $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$ then $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$,

  • (b) $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$ is a proper subset of $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$;

• (2) If $x,y\in l_{\mathrm{\infty }}$ and $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$ then $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$;

• (3) $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y\cap l_{\mathrm{\infty }}=x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y\cap l_{\mathrm{\infty }}$,

where $l_{\mathrm{\infty }}$ denote the set of bounded sequences.

Proof Part (1a): If $ϵ>0$ and $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$ then

$$\begin{array}{rl}\sum _{k\in I_r}|\frac{x_k}{y_k}-L|& \ge \sum _{k\in I_r\mathrm{\&}|\frac{x_k}{y_k}-L|\ge ϵ}|\frac{x_k}{y_k}-L|\\ \ge ϵ\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|\end{array}$$

and so

$$\frac{1}{\epsilon h_r}\sum _{k\in I_r}|\frac{x_k}{y_k}-L|\ge \frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|.$$

Then, for any $\delta >0$,

$$\{r\in \mathbb{N}:\frac{1}{h_r}|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}|\ge \delta \}\subseteq \{r\in N:\frac{1}{h_r}\sum _{k\in I_r}|\frac{x_k}{y_k}-L|\ge ϵ\cdot \delta \}\in \mathcal{I}.$$

Hence, we have $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$.

Part (1b): $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y\subset x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$, let $x=(x_k)$ be defined as follows: $x_k$ to be $1,2,\dots ,[\sqrt{h_r}]$ at the first $[\sqrt{h_r}]$ integers in $I_r$ and zero otherwise. $y_k=1$ for all k. These two satisfy the following $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$, but the following fails $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$.

Part (2): Suppose $x=(x_k)$ and $y=(y_k)$ are in $l_{\mathrm{\infty }}$ and $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$. Then we can assume that

$$|\frac{x_k}{y_k}-L|\le M\text{for all }k.$$

Given $ϵ>0$, we have

$$\begin{array}{rl}\frac{1}{h_r}\sum _{k\in I_r}|\frac{x_k}{y_k}-L|& =\frac{1}{h_r}\sum _{k\in I_r\mathrm{\&}|\frac{x_k}{y_k}-L|\ge ϵ}|\frac{x_k}{y_k}-L|+\frac{1}{h_r}\sum _{k\in I_r\mathrm{\&}|\frac{x_k}{y_k}-L|<ϵ}|\frac{x_k}{y_k}-L|\\ \le \frac{M}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge \frac{ϵ}{2}\}\right|+\frac{ϵ}{2}.\end{array}$$

Consequently, we have

$$\{r\in \mathbb{N}:\frac{1}{h_r}\sum _{k\in I_r}|\frac{x_k}{y_k}-L|\ge \epsilon \}\subseteq \{r\in N:\frac{1}{h_r}|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge \frac{ϵ}{2}\}|\ge \frac{\epsilon }{2M}\}\in \mathcal{I}.$$

Therefore, $x\stackrel{N_{\theta }^L(\mathcal{I})}{\sim }y$.

Part (3): Follows from (1) and (2). □

Theorem 4.2 Let ℐ is an ideal and $\theta =\{k_r\}$ is a lacunary sequence with $liminf{q}_r>1$, then

$$x\stackrel{S^L(\mathcal{I})}{\sim }y\mathit{\text{implies}}x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y.$$

Proof Suppose first that $liminf{q}_r>1$, then there exists a $\delta >0$ such that $q_r\ge 1+\delta$ for sufficiently large r, which implies

$$\frac{h_r}{k_r}\ge \frac{\delta }{1+\delta }.$$

If $x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y$, then for every $\epsilon >0$ and for sufficiently large r, we have

$$\begin{array}{rl}\frac{1}{k_r}\left|\{k\le k_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|& \ge \frac{1}{k_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|\\ \ge \frac{\delta }{1+\delta }\frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|.\end{array}$$

Then, for any $\delta >0$, we get

$$\begin{array}{r}\{r\in \mathbb{N}:\frac{1}{h_r}|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge \epsilon \}|\ge \delta \}\\ \subseteq \{r\in N:\frac{1}{k_r}|\{k\le k_r:|\frac{x_k}{y_k}-L|\ge ϵ\}|\ge \frac{\delta \alpha }{(1+\alpha )}\}\in \mathcal{I}.\end{array}$$

This completes the proof. □

For the next result we assume that the lacunary sequence θ satisfies the condition that for any set $C\in F(\mathcal{I})$, $\bigcup \{n:k_{r-1}<n<k_r,r\in C\}\in F(\mathcal{I})$.

Theorem 4.3 Let ℐ is an ideal and $\theta =(k_r)$ is a lacunary sequence with $sup{q}_r<\mathrm{\infty }$, then

$$x\stackrel{S_{\theta }^L(\mathcal{I})}{\sim }y\mathit{\text{implies}}x\stackrel{S^L(\mathcal{I})}{\sim }y.$$

Proof If ${limsup}_r{q}_r<\mathrm{\infty }$, then without any loss of generality, we can assume that there exists a $0<B<\mathrm{\infty }$ such that $q_r<B$ for all $r\ge 1$. Suppose that $x\stackrel{S_{\theta }^L}{\sim }y$ and for $ϵ,\delta ,{\delta }_1>0$ define the sets

$$C=\{r\in \mathbb{N}:\frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|<\delta \}$$

and

$$T=\{n\in \mathbb{N}:\frac{1}{n}\left|\{k\le n:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|<{\delta }_1\}.$$

It is obvious from our assumption that $C\in F(\mathcal{I})$, the filter associated with the ideal ℐ. Further observe that

$$A_j=\frac{1}{h_j}\left|\{k\in I_j:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|<\delta$$

for all $j\in C$. Let $n\in \mathbb{N}$ be such that $k_{r-1}<n<k_r$ for some $r\in C$. Now

$$\begin{array}{r}\frac{1}{n}\left|\{k\le n:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|\\ \le \frac{1}{k_{r-1}}\left|\{k\le k_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|\\ =\frac{1}{k_{r-1}}\left|\{k\in I_1:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|+\cdots +\frac{1}{k_{r-1}}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|\\ =\frac{k_1}{k_{r-1}}\frac{1}{h_1}\left|\{k\in I_1:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|+\frac{k_2-k_1}{k_{r-1}}\frac{1}{h_2}\left|\{k\in I_2:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|+\cdots \\ +\frac{k_r-k_{r-1}}{k_{r-1}}\frac{1}{h_r}\left|\{k\in I_r:|\frac{x_k}{y_k}-L|\ge ϵ\}\right|\\ =\frac{k_1}{k_{r-1}}{A}_1+\frac{k_2-k_1}{k_{r-1}}{A}_2+\cdots +\frac{k_r-k_{r-1}}{k_{r-1}}{A}_r\\ \le \underset{j\in C}{sup}{A}_j\cdot \frac{k_r}{k_{r-1}}<B\delta .\end{array}$$

Choosing ${\delta }_1=\frac{\delta }{B}$ and in view of the fact that $\bigcup \{n:k_{r-1}<n<k_r,r\in C\}\subset T$ where $C\in F(\mathcal{I})$, it follows from our assumption on θ that the set T also belongs to $F(\mathcal{I})$ and this completes the proof of the theorem. □