Theory and Modern Applications

# Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces

## Abstract

The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and ${S}^{\ast }\left(\tau \right)$-convergence through de la Vallée-Poussin mean in a locally solid Riesz space.

MSC:40A35, 40G15, 46A40.

## 1 Introduction and preliminaries

Since 1951, when Steinhaus [1] and Fast [2] defined statistical convergence for sequences of real numbers, several generalizations and applications of this notion have been investigated. For more detail and related concepts, we refer to [329] and references therein. Quite recently, Di Maio and Kǒcinac [30] studied this notion in topological and uniform spaces and Albayrak and Pehlivan [31], and Mohiuddine and Alghamdi [32] for real and lacunary sequences, respectively, in locally solid Riesz spaces. Afterward, the idea was extended to double sequences by Mohiuddine et al.[33] in the framework of locally solid Riesz spaces.

Let K be a subset of , the set of natural numbers. Then the asymptotic density of K denoted by $\delta \left(K\right)$ is defined as

$\delta \left(K\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}|\left\{k\le n:k\in K\right\}|,$

where the vertical bars denote the cardinality of the enclosed set.

The number sequence $x=\left({x}_{j}\right)$ is said to be statistically convergent to the number if for each $ϵ>0$,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}|\left\{j\le n:|{x}_{j}-\ell |\ge ϵ\right\}|=0.$

In this case, we write $\mathit{st}\text{-}lim{x}_{j}=\ell$.

Remark 1.1 It is well known that every statistically convergent sequence is convergent, but the converse is not true. For example, suppose that the sequence $x=\left({x}_{n}\right)$ is defined as

It is clear that the sequence $x=\left({x}_{n}\right)$ is statistically convergent to 0, but it is not convergent.

Now we recall some definitions related to the notion of a locally solid Riesz space. Let X be a real vector space and ≤ be a partial order on this space. Then X is said to be an ordered vector space if it satisfies the following properties:

1. (i)

If $x,y\in X$ and $y\le x$, then $y+z\le x+z$ for each $z\in X$.

2. (ii)

If $x,y\in X$ and $y\le x$, then $\lambda y\le \lambda x$ for each $\lambda \ge 0$.

If in addition X is a lattice with respect to the partial order ≤, then X is said to be a Riesz space (or a vector lattice) [34].

For an element x of a Riesz space X, the positive part of x is defined by ${x}^{+}=x\vee \theta =sup\left\{x,\theta \right\}$, the negative part of x by ${x}^{-}=\left(-x\right)\vee \theta$ and the absolute value of x by $|x|=x\vee \left(-x\right)$, where θ is the zero element of X.

A subset S of a Riesz space X is said to be solid if $y\in S$ and $|x|\le |y|$ imply $x\in S$.

A topological vector space$\left(X,\tau \right)$ is a vector space X which has a (linear) topology τ such that the algebraic operations of addition and scalar multiplication in X are continuous. The continuity of addition means that the function $f:X×X\to X$ defined by $f\left(x,y\right)=x+y$ is continuous on $X×X$, and the continuity of scalar multiplication means that the function $f:\mathbb{R}×X\to X$ defined by $f\left(\lambda ,x\right)=\lambda x$ is continuous on $\mathbb{R}×X$.

Every linear topology τ on a vector space X has a base $\mathcal{N}$ for the neighborhoods of θ satisfying the following properties:

(C1) Each $Y\in \mathcal{N}$ is a balanced set, that is, $\lambda x\in Y$ holds for all $x\in Y$ and every $\lambda \in \mathbb{R}$ with $|\lambda |\le 1$.

(C2) Each $Y\in \mathcal{N}$ is an absorbing set, that is, for every $x\in X$, there exists $\lambda >0$ such that $\lambda x\in Y$.

(C3) For each $Y\in \mathcal{N}$, there exists some $E\in \mathcal{N}$ with $E+E\subseteq Y$.

A linear topology τ on a Riesz space X is said to be locally solid (cf.[35, 36]) if τ has a base at zero consisting of solid sets. A locally solid Riesz space$\left(X,\tau \right)$ is a Riesz space equipped with a locally solid topology τ.

In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and ${S}^{\ast }\left(\tau \right)$-convergence through de la Vallée-Poussin mean in a locally solid Riesz space.

## 2 Generalized statistical τ-convergence

Throughout the text, we write ${\mathcal{N}}_{\mathrm{sol}}$ for any base at zero consisting of solid sets and satisfying the conditions (C1), (C2) and (C3) in a locally solid topology. The following idea of λ-statistical convergence was introduced in [37] and further studied in [3840].

Let $\lambda =\left({\lambda }_{n}\right)$ be a non-decreasing sequence of positive numbers tending to ∞ such that

${\lambda }_{n+1}\le {\lambda }_{n}+1,\phantom{\rule{2em}{0ex}}{\lambda }_{1}=0.$

The generalized de la Vallée-Poussin mean is defined by

${t}_{n}\left(x\right)=:\frac{1}{{\lambda }_{n}}\sum _{j\in {I}_{n}}{x}_{j},$

where ${I}_{n}=\left[n-{\lambda }_{n}+1,n\right]$.

A sequence $x=\left({x}_{j}\right)$ is said to be $\left(V,\lambda \right)$-summable to a number if

A sequence $x=\left({x}_{j}\right)$ is said to be strongly$\left(V,\lambda \right)$-summable to a number if

We denote it by ${x}_{j}\to \ell \left[V,\lambda \right]$ as $j\to \mathrm{\infty }$.

Let $K\subseteq \mathbb{N}$ be a set of positive integers, then

${\delta }_{\lambda }\left(K\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{n-{\lambda }_{n}+1\le j\le n:j\in K\right\}|$

is said to be the λ-density of K.

In case ${\lambda }_{n}=n$, the λ-density reduces to the natural density.

The number sequence $x=\left({x}_{j}\right)$ is said to be λ-statistically convergent to the number if for each $ϵ>0$, ${\delta }_{\lambda }\left({K}_{ϵ}\right)=0$, where ${K}_{ϵ}=\left\{j\in \mathbb{N}:|{x}_{j}-\ell |>ϵ\right\}$, i.e.,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:|{x}_{j}-\ell |>ϵ\right\}|=0.$

In this case, we write ${\mathit{st}}_{\lambda }\text{-}{lim}_{j}{x}_{j}=\ell$ and we denote the set of all λ-statistically convergent sequences by ${S}_{\lambda }$. This notion was extended to double sequences in [41, 42].

Remark 2.1 As in Remark 1.1, we observe that if a sequence is $\left(V,\lambda \right)$-summable to a number , then it is also λ-statistically convergent to the same number , but the converse need not be true. For example, let the sequence $z=\left({z}_{k}\right)$ be defined by

where $\left[a\right]$ denotes the integer part of $a\in \mathbb{R}$. Then x is λ-statistically convergent to 0 but not $\left(V,\lambda \right)$-summable.

Definition 2.1 Let $\left(X,\tau \right)$ be a locally solid Riesz space. Then a sequence $x=\left({x}_{j}\right)$ in X is said to be generalized statistically τ-convergent (or${S}_{\lambda }\left(\tau \right)$-convergent) to the number $\xi \in X$ if for every τ-neighborhood U of zero,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:{x}_{j}-\xi \notin U\right\}|=0.$

In this case, we write ${S}_{\lambda }\left(\tau \right)\text{-}limx=\xi$ or ${x}_{j}\stackrel{{S}_{\lambda }\left(\tau \right)}{⟶}\xi$.

Definition 2.2 Let $\left(X,\tau \right)$ be a locally solid Riesz space. We say that a sequence $x=\left({x}_{j}\right)$ in X is generalized statistically τ-bounded if for every τ-neighborhood U of zero, there exists some $\lambda >0$ such that the set

$\left\{j\in \mathbb{N}:\lambda {x}_{j}\notin U\right\}$

has λ-density zero.

Theorem 2.1 Let$\left(X,\tau \right)$be a Hausdorff locally solid Riesz space and$x=\left({x}_{j}\right)$and$y=\left({y}_{k}\right)$be two sequences in X. Then the following hold:

1. (i)

If ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}={\xi }_{1}$ and ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}={\xi }_{2}$, then ${\xi }_{1}={\xi }_{2}$.

2. (ii)

If ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi$, then ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}\alpha {x}_{j}=\alpha \xi$, $\alpha \in \mathbb{R}$.

3. (iii)

If ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi$ and ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{y}_{j}=\eta$, then ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}\left({x}_{j}+{y}_{j}\right)=\xi +\eta$.

Proof (i) Suppose that ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}={\xi }_{1}$ and ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}={\xi }_{2}$. Let U be any τ-neighborhood of zero. Then there exists $Y\in {\mathcal{N}}_{\mathrm{sol}}$ such that $Y\subseteq U$. Choose any $E\in {\mathcal{N}}_{\mathrm{sol}}$ such that $E+E\subseteq Y$. We define the following sets:

$\begin{array}{c}{K}_{1}=\left\{j\in \mathbb{N}:{x}_{j}-{\xi }_{1}\in E\right\},\hfill \\ {K}_{2}=\left\{j\in \mathbb{N}:{x}_{j}-{\xi }_{2}\in E\right\}.\hfill \end{array}$

Since ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}={\xi }_{1}$ and ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}={\xi }_{2}$, we have ${\delta }_{\lambda }\left({K}_{1}\right)={\delta }_{\lambda }\left({K}_{2}\right)=1$. Thus $\delta \left({K}_{1}\cap {K}_{2}\right)=1$ and, in particular, ${K}_{1}\cap {K}_{2}\ne \mathrm{\varnothing }$. Now, let $j\in {K}_{1}\cap {K}_{2}$. Then

${\xi }_{1}-{\xi }_{2}={\xi }_{1}-{x}_{j}+{x}_{j}-{\xi }_{2}\in E+E\subseteq Y\subseteq U.$

Hence, for every τ-neighborhood U of zero, we have ${\xi }_{1}-{\xi }_{2}\in U$. Since $\left(X,\tau \right)$ is Hausdorff, the intersection of all τ-neighborhoods U of zero is the singleton set $\left\{\theta \right\}$. Thus, we get ${\xi }_{1}-{\xi }_{2}=\theta$, i.e., ${\xi }_{1}={\xi }_{2}$.

1. (ii)

Let U be an arbitrary τ-neighborhood of zero and ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi$. Then there exists $Y\in {\mathcal{N}}_{\mathrm{sol}}$ such that $Y\subseteq U$ and also

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:{x}_{j}-\xi \in Y\right\}|=1.$

Since Y is balanced, ${x}_{j}-\xi \in Y$ implies $\alpha \left({x}_{j}-\xi \right)\in Y$ for every $\alpha \in \mathbb{R}$ with $|\alpha |\le 1$. Hence, for every $n\in \mathbb{N}$, we get

$\begin{array}{rcl}\left\{j\in {I}_{n}:{x}_{j}-\xi \in Y\right\}& \subseteq & \left\{j\in {I}_{n}:\alpha {x}_{j}-\alpha \xi \in Y\right\}\\ \subseteq & \left\{j\in {I}_{n}:\alpha {x}_{j}-\alpha \xi \in U\right\}.\end{array}$

Thus, we obtain

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:\alpha {x}_{j}-\alpha \xi \in U\right\}|=1$

for each τ-neighborhood U of zero. Now let $|\alpha |>1$ and $\left[|\alpha |\right]$ be the smallest integer greater than or equal to $|\alpha |$. There exists $E\in {\mathcal{N}}_{\mathrm{sol}}$ such that $\left[|\alpha |\right]E\subseteq Y$. Since ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi$, the set

$K=\left\{j\in \mathbb{N}:{x}_{j}-\xi \in E\right\}$

has λ-density zero. Therefore, for all $n\in \mathbb{N}$ and $j\in K\cap {I}_{n}$, we have

$|\alpha \xi -\alpha {x}_{j}|=|\alpha ||\xi -{x}_{j}|\le \left[|\alpha |\right]|\xi -{x}_{j}|\in \left[|\alpha |\right]E\subseteq Y\subseteq U.$

Since the set Y is solid, we have $\alpha \xi -\alpha {x}_{j}\in Y$. This implies that $\alpha \xi -\alpha {x}_{j}\in U$. Thus,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:\alpha {x}_{j}-\alpha \xi \in U\right\}|=1$

for each τ-neighborhood U of zero. Hence ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}\alpha {x}_{j}=\alpha \xi$.

1. (iii)

Let U be an arbitrary τ-neighborhood of zero. Then there exists $Y\in {\mathcal{N}}_{\mathrm{sol}}$ such that $Y\subseteq U$. Choose E in ${\mathcal{N}}_{\mathrm{sol}}$ such that $E+E\subseteq Y$. Since ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi$ and ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{y}_{j}=\eta$, we have ${\delta }_{\lambda }\left({H}_{1}\right)=1={\delta }_{\lambda }\left({H}_{2}\right)$, where

$\begin{array}{c}{H}_{1}=\left\{j\in \mathbb{N}:{x}_{j}-\xi \in E\right\},\hfill \\ {H}_{2}=\left\{j\in \mathbb{N}:{y}_{j}-\eta \in E\right\}.\hfill \end{array}$

Let $H={H}_{1}\cap {H}_{2}$. Hence, we have ${\delta }_{\lambda }\left(H\right)=1$. For all $n\in \mathbb{N}$ and $j\in H\cap {I}_{n}$, we get

$\left({x}_{j}+{y}_{j}\right)-\left(\xi +\eta \right)=\left({x}_{j}-\xi \right)+\left({y}_{j}-\eta \right)\in E+E\subseteq Y\subseteq U.$

Therefore,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:\left({x}_{j}+{y}_{j}\right)-\left(\xi +\eta \right)\in U\right\}|=1.$

Since U is arbitrary, we have ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}\left({x}_{j}+{y}_{j}\right)=\xi +\eta$. □

Theorem 2.2 Let$\left(X,\tau \right)$be a locally solid Riesz space. If a sequence$x=\left({x}_{j}\right)$is generalized statistically τ-convergent, then it is generalized statistically τ-bounded.

Proof Suppose $x=\left({x}_{j}\right)$ is generalized statistically τ-convergent to the point $\xi \in X$ and let U be an arbitrary τ-neighborhood of zero. Then there exists $Y\in {\mathcal{N}}_{\mathrm{sol}}$ such that $Y\subseteq U$. Let us choose $E\in {\mathcal{N}}_{\mathrm{sol}}$ such that $E+E\subseteq Y$. Since ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j\to \mathrm{\infty }}{x}_{j}=\xi$, the set

$K=\left\{j\in \mathbb{N}:{x}_{j}-\xi \notin E\right\}$

has λ-density zero. Since E is absorbing, there exists $\lambda >0$ such that $\lambda \xi \in E$. Let $\alpha \in \left(0,min\left\{1,\lambda \right\}\right)$. Since E is solid and $|\alpha \xi |\le |\lambda x|$, we have $\alpha \xi \in E$. Since E is balanced, ${x}_{j}-\xi \in E$ implies $\alpha \left({x}_{j}-\xi \right)\in E$. Then, for each $n\in \mathbb{N}$ and $j\in \left(\mathbb{N}\setminus K\right)\cap {I}_{n}$, we have

$\alpha {x}_{j}=\alpha \left({x}_{j}-\xi \right)+\alpha \xi \in E+E\subseteq Y\subseteq U.$

Thus

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:\alpha {x}_{j}\notin U\right\}|=0.$

Hence, $\left({x}_{j}\right)$ is generalized statistically τ-bounded. □

Theorem 2.3 Let$\left(X,\tau \right)$be a locally solid Riesz space. If$\left({x}_{j}\right)$, $\left({y}_{j}\right)$and$\left({z}_{j}\right)$are three sequences such that

1. (i)

${x}_{j}\le {y}_{j}\le {z}_{j}$ for all $j\in \mathbb{N}$,

2. (ii)

${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi ={S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{z}_{j}$,

then${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{y}_{j}=\xi$.

Proof Let U be an arbitrary τ-neighborhood of zero, there exists $Y\in {\mathcal{N}}_{\mathrm{sol}}$ such that $Y\subseteq U$. Choose $E\in {\mathcal{N}}_{\mathrm{sol}}$ such that $E+E\subseteq Y$. From condition (ii), we have ${\delta }_{\lambda }\left(A\right)=1={\delta }_{\lambda }\left(B\right)$, where

$\begin{array}{c}A=\left\{j\in \mathbb{N}:{x}_{j}-\xi \in E\right\},\hfill \\ B=\left\{j\in \mathbb{N}:{x}_{j}-\xi \in E\right\}.\hfill \end{array}$

Also, we get ${\delta }_{\lambda }\left(A\cap B\right)=1$, and from (i) we have

${x}_{j}-\xi \le {y}_{j}-\xi \le {z}_{j}-\xi$

for all $j\in \mathbb{N}$. This implies that for all $n\in \mathbb{N}$ and $j\in A\cap B\cap {I}_{n}$, we get

$|{y}_{j}-\xi |\le |{x}_{j}-\xi |+|{z}_{j}-\xi |\in E+E\subseteq Y.$

Since Y is solid, we have ${y}_{j}-\xi \in Y\subseteq U$. Thus,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:{y}_{j}-\xi \in U\right\}|=1$

for each τ-neighborhood U of zero. Hence ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{y}_{j}=\xi$. □

## 3 Generalized statistically τ-Cauchy and ${S}_{\lambda }^{\ast }\left(\tau \right)$-convergence

Definition 3.1 Let $\left(X,\tau \right)$ be a locally solid Riesz space. A sequence $x=\left({x}_{j}\right)$ in X is generalized statistically τ-Cauchy if for every τ-neighborhood U of zero there exists $p\in \mathbb{N}$ such that the set

$\left\{j\in \mathbb{N}:{x}_{j}-{x}_{p}\notin U\right\}$

has λ-density zero.

Theorem 3.1 Let$\left(X,\tau \right)$be a locally solid Riesz space. If a sequence$x=\left({x}_{j}\right)$is generalized statistically τ-convergent, then it is generalized statistically τ-Cauchy.

Proof Suppose that ${S}_{\lambda }\left(\tau \right)\text{-}{lim}_{j}{x}_{j}=\xi$. Let U be an arbitrary τ-neighborhood of zero, there exists $Y\in {\mathcal{N}}_{\mathrm{sol}}$ such that $Y\subseteq U$. Choose $E\in {\mathcal{N}}_{\mathrm{sol}}$ such that $E+E\subseteq Y$. By generalized statistical τ-convergence to ξ, there is $p\in \mathbb{N}$ with $\xi -{x}_{p}\in E$ and

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_{n}}|\left\{j\in {I}_{n}:{x}_{j}-\xi \notin E\right\}|=0.$

Also, for all $n\in \mathbb{N}$ and $j\in \left(\mathbb{N}\setminus K\right)\cap {I}_{n}$, where

$K=\left\{j\in \mathbb{N}:{x}_{j}-\xi \notin E\right\},$

we have

${x}_{j}-{x}_{p}={x}_{j}-\xi +\xi -{x}_{p}\in E+E\subseteq Y\subseteq U$

and ${\delta }_{\lambda }\left(K\right)=0$. Therefore the set

$\left\{j\in \mathbb{N}:{x}_{j}-{x}_{p}\notin U\right\}\subseteq K\cap {I}_{n}$

for all $n\in \mathbb{N}$. For every τ-neighborhood U of zero there exists $p\in \mathbb{N}$ such that the set $\left\{j\in \mathbb{N}:{x}_{j}-{x}_{p}\notin U\right\}$ has λ-density zero. Hence $\left({x}_{j}\right)$ is generalized statistically τ-Cauchy. □

Now we define another type of convergence in locally solid Riesz spaces.

Definition 3.2 A sequence $\left({x}_{j}\right)$ in a locally solid Riesz space $\left(X,\tau \right)$ is said to be ${S}_{\lambda }^{\ast }\left(\tau \right)$-convergent to $\xi \in X$ if there exists an index set $K=\left\{{j}_{n}\right\}\subseteq \mathbb{N}$, $n=1,2,\dots$ , with ${\delta }_{\lambda }\left(K\right)=1$ such that ${lim}_{n\to \mathrm{\infty }}{x}_{{j}_{n}}=\xi$. In this case, we write $\xi ={S}_{\lambda }^{\ast }\left(\tau \right)\text{-}limx$.

Theorem 3.2 A sequence$x=\left({x}_{j}\right)$in a locally solid Riesz space$\left(X,\tau \right)$is generalized statistically τ-convergent to a number ξ if it is${S}_{\lambda }^{\ast }\left(\tau \right)$-convergent to ξ.

Proof Let U be an arbitrary τ-neighborhood of ξ. Since $x=\left({x}_{j}\right)$ is ${S}_{\lambda }^{\ast }\left(\tau \right)$-convergent to ξ, there is an index set $K=\left\{{j}_{n}\right\}\subseteq \mathbb{N}$, $n=1,2,\dots$ , with ${\delta }_{\lambda }\left(K\right)=1$ and ${j}_{0}={j}_{0}\left(U\right)$, such that $j\ge {j}_{0}$ and $j\in K$ imply ${x}_{j}-\xi \in U$. Then

${K}_{U}=\left\{j\in \mathbb{N}:{x}_{j}-\xi \notin U\right\}\subseteq \mathbb{N}-\left\{{j}_{N+1},{j}_{N+2},\dots \right\}.$

Therefore ${\delta }_{\lambda }\left({K}_{U}\right)=0$. Hence x is generalized statistically τ-convergent to ξ. □

Note that the converse holds for a first countable space.

Recall that a topological space is first countable if each point has a countable (decreasing) local base.

Theorem 3.3 Let$\left(X,\tau \right)$be a first countable locally solid Riesz space. If a sequence$x=\left({x}_{j}\right)$is generalized statistically τ-convergent to a number ξ, then it is${S}_{\lambda }^{\ast }\left(\tau \right)$-convergent to ξ.

Proof Let x be generalized statistically τ-convergent to a number ξ. Fix a countable local base ${U}_{1}\supset {U}_{2}\supset {U}_{3}\supset \cdots$ at ξ. For each $i\in \mathbb{N}$, put

${K}_{i}=\left\{j\in \mathbb{N}:{x}_{j}-\xi \notin {U}_{i}\right\}.$

By hypothesis, ${\delta }_{\lambda }\left({K}_{i}\right)=0$ for each i. Since the ideal of all subsets of having λ-density zero is a P-ideal (see, for instance, [43]), then there exists a sequence of sets ${\left({J}_{i}\right)}_{i}$ such that the symmetric difference ${K}_{i}\mathrm{\Delta }{J}_{i}$ is a finite set for any $i\in \mathbb{N}$ and $J:={\bigcup }_{i=1}^{\mathrm{\infty }}{J}_{i}\in \mathcal{I}$.

Let $K=\mathbb{N}\setminus J$, then ${\delta }_{\lambda }\left(K\right)=1$. In order to prove the theorem, it is enough to check that ${lim}_{j\in K}{x}_{j}=\xi$.

Let $i\in \mathbb{N}$. Since ${K}_{i}\mathrm{\Delta }{J}_{i}$ is finite, there is ${j}_{i}\in \mathbb{N}$, without loss of generality, with ${j}_{i}\in K$, ${j}_{i}>i$, such that

$\left(\mathbb{N}\setminus {J}_{i}\right)\cap \left\{j\in \mathbb{N}:j\ge {j}_{i}\right\}=\left(\mathbb{N}\setminus {K}_{i}\right)\cap \left\{j\in \mathbb{N}:j\ge {j}_{i}\right\}.$
(1)

If $j\in K$ and $j\ge {j}_{i}$, then $j\notin {J}_{i}$, and by (1), $j\notin {K}_{i}$. Thus ${x}_{j}-\xi \in {U}_{i}$. So, we have proved that for all $i\in \mathbb{N}$, there is ${j}_{i}\in K$, ${j}_{i}>i$, with ${x}_{j}-\xi \in {U}_{i}$ for every $j\ge {j}_{i}$: without loss of generality, we can suppose ${j}_{i+1}>{j}_{i}$ for every $i\in \mathbb{N}$. The assertion follows taking into account that the ${U}_{i}$’s form a countable local base at ξ. □

## 4 Conclusion

Recently, statistical convergence has been established as a better option than ordinary convergence. It is found very interesting that some results on sequences, series and summability can be proved by replacing the ordinary convergence by statistical convergence; and further, through some examples, where some efforts are required, we can show that the results for statistical convergence happen to be stronger than those proved for ordinary convergence (e.g., [4449]). This notion has also been defined and studied in different setups. In this paper, we have studied this notion through de la Vallée-Poussin mean in a locally solid Riesz space to deal with the convergence problems in a broader sense.

## References

1. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73-74.

2. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241-244.

3. Çakalli H: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 1995, 26(2):113-119.

4. Çakalli H: On statistical convergence in topological groups. Pure Appl. Math. Sci. 1996, 43: 27-31.

5. Çakalli H, Khan MK: Summability in topological spaces. Appl. Math. Lett. 2011, 24: 348-352. 10.1016/j.aml.2010.10.021

6. Çakalli H, Savaş E: Statistical convergence of double sequence in topological groups. J. Comput. Anal. Appl. 2010, 12(2):421-426.

7. Edely OHH, Mursaleen M: On statistical A -summability. Math. Comput. Model. 2009, 49: 672-680. 10.1016/j.mcm.2008.05.053

8. Fridy JA: On statistical convergence. Analysis 1985, 5: 301-313.

9. Karakuş S, Demirci K: Statistical convergence of double sequences on probabilistic normed spaces. Int. J. Math. Math. Sci. 2007., 2007: Article ID 14737

10. Karakuş S, Demirci K, Duman O: Statistical convergence on intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2008, 35: 763-769. 10.1016/j.chaos.2006.05.046

11. Maddox IJ: Statistical convergence in a locally convex space. Math. Proc. Camb. Philos. Soc. 1988, 104: 141-145. 10.1017/S0305004100065312

12. Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inf. Sci. 2012, 6(3):581-585.

13. Mohuiddine SA, Alotaibi A, Alsulami SM: Ideal convergence of double sequences in random 2-normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 149

14. Mohiuddine SA, Danish Lohani QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 1731-1737. 10.1016/j.chaos.2009.03.086

15. Mohiuddine SA, Savaş E: Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara 2012, 58: 331-339. 10.1007/s11565-012-0157-5

16. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl. 2010, 12(4):787-798.

17. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence of double sequences in fuzzy normed spaces. Filomat 2012, 26(4):673-681. 10.2298/FIL1204673M

18. Mursaleen M: On statistical convergence in random 2-normed spaces. Acta Sci. Math. 2010, 76: 101-109.

19. Mursaleen M, Alotaibi A: On I -convergence in random 2-normed spaces. Math. Slovaca 2011, 61(6):933-940. 10.2478/s12175-011-0059-5

20. Mursaleen M, Edely OHH: Generalized statistical convergence. Inf. Sci. 2004, 162: 287-294. 10.1016/j.ins.2003.09.011

21. Mursaleen M, Edely OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 2003, 288: 223-231. 10.1016/j.jmaa.2003.08.004

22. Mursaleen M, Edely OHH: On the invariant mean and statistical convergence. Appl. Math. Lett. 2009, 22: 1700-1704. 10.1016/j.aml.2009.06.005

23. Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41: 2414-2421. 10.1016/j.chaos.2008.09.018

24. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142-149. 10.1016/j.cam.2009.07.005

25. Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces. Math. Rep. 2010, 12(64)(4):359-371.

26. Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca 2012, 62: 49-62. 10.2478/s12175-011-0071-9

27. Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl. 2010, 59: 603-611. 10.1016/j.camwa.2009.11.002

28. Savaş E, Mohiuddine SA:$\overline{\lambda }$-statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012, 62(1):99-108. 10.2478/s12175-011-0075-5

29. Savaş E, Mursaleen M: On statistically convergent double sequences of fuzzy numbers. Inf. Sci. 2004, 162: 183-192. 10.1016/j.ins.2003.09.005

30. Di Maio G, Kočinac LDR: Statistical convergence in topology. Topol. Appl. 2008, 156: 28-45. 10.1016/j.topol.2008.01.015

31. Albayrak H, Pehlivan S: Statistical convergence and statistical continuity on locally solid Riesz spaces. Topol. Appl. 2012, 159: 1887-1893. 10.1016/j.topol.2011.04.026

32. Mohiuddine SA, Alghamdi MA: Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl. 2012., 2012: Article ID 225

33. Mohiuddine SA, Alotaibi A, Mursaleen M: Statistical convergence of double sequences in locally solid Riesz spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 719729

34. Zaanen AC: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin; 1997.

35. Aliprantis CD, Burkinshaw O: Locally Solid Riesz Spaces with Applications to Economics. 2nd edition. Am. Math. Soc., Providence; 2003.

36. Roberts GT: Topologies in vector lattices. Math. Proc. Camb. Philos. Soc. 1952, 48: 533-546. 10.1017/S0305004100076295

37. Mursaleen M: λ -statistical convergence. Math. Slovaca 2000, 50: 111-115.

38. Çolak R, Bektaş CA: λ -statistical convergence of order α . Acta Math. Sci., Ser. B 2011, 31(3):953-959.

39. Edely OHH, Mohiuddine SA, Noman AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 2010, 23: 1382-1387. 10.1016/j.aml.2010.07.004

40. de Malafosse B, Rakočević V: Matrix transformation and statistical convergence. Linear Algebra Appl. 2007, 420: 377-387. 10.1016/j.laa.2006.07.021

41. Mursaleen M, Çakan C, Mohiuddine SA, Savaş E: Generalized statistical convergence and statistical core of double sequences. Acta Math. Sin. Engl. Ser. 2010, 26: 2131-2144. 10.1007/s10114-010-9050-2

42. Kumar V, Mursaleen M:On $\left(\lambda ,\mu \right)$-statistical convergence of double sequences on intuitionistic fuzzy normed spaces. Filomat 2011, 25(2):109-120. 10.2298/FIL1102109K

43. Farah I Mem. Amer. Math. Soc. 148. Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers 2000.

44. Caserta A, Kočinac LDR: On statistical exhaustiveness. Appl. Math. Lett. 2012, 25: 1447-1451. 10.1016/j.aml.2011.12.022

45. Caserta A, Di Maio G, Kočinac LDR: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 420419

46. Mohiuddine SA: An application of almost convergence in approximation theorems. Appl. Math. Lett. 2011, 24: 1856-1860. 10.1016/j.aml.2011.05.006

47. Mohiuddine SA, Alotaibi A: Statistical convergence and approximation theorems for functions of two variables. J. Comput. Anal. Appl. 2013, 15(2):218-223.

48. Mohiuddine SA, Alotaibi A, Mursaleen M:Statistical summability $\left(C,1\right)$ and a Korovkin type approximation theorem. J. Inequal. Appl. 2012., 2012: Article ID 172

49. Srivastava HM, Mursaleen M, Khan A: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model. 2012, 55: 2040-2051. 10.1016/j.mcm.2011.12.011

## Acknowledgements

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

## Author information

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Correspondence to Syed Abdul Mohiuddine.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Mohiuddine, S.A., Alotaibi, A. & Mursaleen, M. Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces. Adv Differ Equ 2013, 66 (2013). https://doi.org/10.1186/1687-1847-2013-66