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 $S^{\ast }(\tau )$-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 [3–29] 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 ℕ, the set of natural numbers. Then the asymptotic density of K denoted by $\delta (K)$ is defined as

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

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

The number sequence $x=(x_j)$ is said to be statistically convergent to the number ℓ 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 $x=(x_n)$ is defined as

$$x=(x_n)=\{\begin{array}{cc}\sqrt{n}\hfill & \text{if }n\text{ is a square},\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

It is clear that the sequence $x=(x_n)$ 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:

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

• (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 $x^{+}=x\vee \theta =sup\{x,\theta \}$, the negative part of x by $x^{-}=(-x)\vee \theta$ and the absolute value of x by $|x|=x\vee (-x)$, 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$(X,\tau )$ 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 $f:X\times X\to X$ defined by $f(x,y)=x+y$ is continuous on $X\times X$, and the continuity of scalar multiplication means that the function $f:\mathbb{R}\times X\to X$ defined by $f(\lambda ,x)=\lambda x$ is continuous on $\mathbb{R}\times X$.

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$(X,\tau )$ 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 $S^{\ast }(\tau )$-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 [38–40].

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

$${\lambda }_{n+1}\le {\lambda }_n+1,{\lambda }_1=0.$$

The generalized de la Vallée-Poussin mean is defined by

$$t_n(x)=:\frac{1}{{\lambda }_n}\sum _{j\in I_n}{x}_j,$$

where $I_n=[n-{\lambda }_n+1,n]$.

A sequence $x=(x_j)$ is said to be $(V,\lambda )$-summable to a number ℓ if

$$t_n(x)\to \ell \text{as }n\to \mathrm{\infty }.$$

A sequence $x=(x_j)$ is said to be strongly$(V,\lambda )$-summable to a number ℓ if

$$\frac{1}{{\lambda }_n}\sum _{j\in I_n}|x_j-\ell |\to 0\text{as }n\to \mathrm{\infty }.$$

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

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

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

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 $x=(x_j)$ is said to be λ-statistically convergent to the number ℓ if for each $ϵ>0$, ${\delta }_{\lambda }(K_{ϵ})=0$, where $K_{ϵ}=\{j\in \mathbb{N}:|x_j-\ell |>ϵ\}$, 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 $(V,\lambda )$-summable to a number ℓ, then it is also λ-statistically convergent to the same number ℓ, but the converse need not be true. For example, let the sequence $z=(z_k)$ be defined by

$$z_k=\{\begin{array}{cc}k\hfill & \text{if }n-[\sqrt{{\lambda }_n}]+1\le k\le n,\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}$$

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

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

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:x_j-\xi \notin U\}|=0.$$

In this case, we write $S_{\lambda }(\tau )\text{-}limx=\xi$ or $x_j\stackrel{S_{\lambda }(\tau )}{⟶}\xi$.

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

$$\{j\in \mathbb{N}:\lambda x_j\notin U\}$$

has λ-density zero.

Theorem 2.1 Let$(X,\tau )$be a Hausdorff locally solid Riesz space and$x=(x_j)$and$y=(y_k)$be two sequences in X. Then the following hold:

• (i) If $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j={\xi }_1$ and $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j={\xi }_2$, then ${\xi }_1={\xi }_2$.

• (ii) If $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j=\xi$, then $S_{\lambda }(\tau )\text{-}{lim}_j\alpha x_j=\alpha \xi$, $\alpha \in \mathbb{R}$.

• (iii) If $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j=\xi$ and $S_{\lambda }(\tau )\text{-}{lim}_j{y}_j=\eta$, then $S_{\lambda }(\tau )\text{-}{lim}_j(x_j+y_j)=\xi +\eta$.

Proof (i) Suppose that $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j={\xi }_1$ and $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j={\xi }_2$. 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:

$$\begin{array}{c}{K}_1=\{j\in \mathbb{N}:x_j-{\xi }_1\in E\},\hfill \\ K_2=\{j\in \mathbb{N}:x_j-{\xi }_2\in E\}.\hfill \end{array}$$

Since $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j={\xi }_1$ and $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j={\xi }_2$, we have ${\delta }_{\lambda }(K_1)={\delta }_{\lambda }(K_2)=1$. Thus $\delta (K_1\cap K_2)=1$ 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 $(X,\tau )$ is Hausdorff, the intersection of all τ-neighborhoods U of zero is the singleton set $\{\theta \}$. Thus, we get ${\xi }_1-{\xi }_2=\theta$, i.e., ${\xi }_1={\xi }_2$.

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

  $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:x_j-\xi \in Y\}|=1.$$

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

$$\begin{array}{rcl}\{j\in I_n:x_j-\xi \in Y\}& \subseteq & \{j\in I_n:\alpha x_j-\alpha \xi \in Y\}\\ \subseteq & \{j\in I_n:\alpha x_j-\alpha \xi \in U\}.\end{array}$$

Thus, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:\alpha x_j-\alpha \xi \in U\}|=1$$

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

$$K=\{j\in \mathbb{N}:x_j-\xi \in E\}$$

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

$$|\alpha \xi -\alpha x_j|=|\alpha ||\xi -x_j|\le [|\alpha |]|\xi -x_j|\in [|\alpha |]E\subseteq Y\subseteq U.$$

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,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:\alpha x_j-\alpha \xi \in U\}|=1$$

for each τ-neighborhood U of zero. Hence $S_{\lambda }(\tau )\text{-}{lim}_j\alpha x_j=\alpha \xi$.

• (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 $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j=\xi$ and $S_{\lambda }(\tau )\text{-}{lim}_j{y}_j=\eta$, we have ${\delta }_{\lambda }(H_1)=1={\delta }_{\lambda }(H_2)$, where

  $$\begin{array}{c}{H}_1=\{j\in \mathbb{N}:x_j-\xi \in E\},\hfill \\ H_2=\{j\in \mathbb{N}:y_j-\eta \in E\}.\hfill \end{array}$$

Let $H=H_1\cap H_2$. Hence, we have ${\delta }_{\lambda }(H)=1$. For all $n\in \mathbb{N}$ and $j\in H\cap I_n$, we get

$$(x_j+y_j)-(\xi +\eta )=(x_j-\xi )+(y_j-\eta )\in E+E\subseteq Y\subseteq U.$$

Therefore,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}\left|\{j\in I_n:(x_j+y_j)-(\xi +\eta )\in U\}\right|=1.$$

Since U is arbitrary, we have $S_{\lambda }(\tau )\text{-}{lim}_j(x_j+y_j)=\xi +\eta$. □

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

Proof Suppose $x=(x_j)$ 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 $S_{\lambda }(\tau )\text{-}{lim}_{j\to \mathrm{\infty }}{x}_j=\xi$, the set

$$K=\{j\in \mathbb{N}:x_j-\xi \notin E\}$$

has λ-density zero. Since E is absorbing, there exists $\lambda >0$ such that $\lambda \xi \in E$. Let $\alpha \in (0,min\{1,\lambda \})$. 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 $\alpha (x_j-\xi )\in E$. Then, for each $n\in \mathbb{N}$ and $j\in (\mathbb{N}\setminus K)\cap I_n$, we have

$$\alpha x_j=\alpha (x_j-\xi )+\alpha \xi \in E+E\subseteq Y\subseteq U.$$

Thus

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:\alpha x_j\notin U\}|=0.$$

Hence, $(x_j)$ is generalized statistically τ-bounded. □

Theorem 2.3 Let$(X,\tau )$be a locally solid Riesz space. If$(x_j)$, $(y_j)$and$(z_j)$are three sequences such that

• (i) $x_j\le y_j\le z_j$ for all $j\in \mathbb{N}$,

• (ii) $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j=\xi =S_{\lambda }(\tau )\text{-}{lim}_j{z}_j$,

then$S_{\lambda }(\tau )\text{-}{lim}_j{y}_j=\xi$.

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 ${\delta }_{\lambda }(A)=1={\delta }_{\lambda }(B)$, where

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

Also, we get ${\delta }_{\lambda }(A\cap B)=1$, 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,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:y_j-\xi \in U\}|=1$$

for each τ-neighborhood U of zero. Hence $S_{\lambda }(\tau )\text{-}{lim}_j{y}_j=\xi$. □

3 Generalized statistically τ-Cauchy and $S_{\lambda }^{\ast }(\tau )$-convergence

Definition 3.1 Let $(X,\tau )$ be a locally solid Riesz space. A sequence $x=(x_j)$ 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

$$\{j\in \mathbb{N}:x_j-x_p\notin U\}$$

has λ-density zero.

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

Proof Suppose that $S_{\lambda }(\tau )\text{-}{lim}_j{x}_j=\xi$. 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

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }_n}|\{j\in I_n:x_j-\xi \notin E\}|=0.$$

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

$$K=\{j\in \mathbb{N}:x_j-\xi \notin E\},$$

we have

$$x_j-x_p=x_j-\xi +\xi -x_p\in E+E\subseteq Y\subseteq U$$

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

$$\{j\in \mathbb{N}:x_j-x_p\notin U\}\subseteq K\cap I_n$$

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

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

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

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

Proof Let U be an arbitrary τ-neighborhood of ξ. Since $x=(x_j)$ is $S_{\lambda }^{\ast }(\tau )$-convergent to ξ, there is an index set $K=\{j_n\}\subseteq \mathbb{N}$, $n=1,2,\dots$ , with ${\delta }_{\lambda }(K)=1$ and $j_0=j_0(U)$, such that $j\ge j_0$ and $j\in K$ imply $x_j-\xi \in U$. Then

$$K_U=\{j\in \mathbb{N}:x_j-\xi \notin U\}\subseteq \mathbb{N}-\{j_{N+1},j_{N+2},\dots \}.$$

Therefore ${\delta }_{\lambda }(K_U)=0$. 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$(X,\tau )$be a first countable locally solid Riesz space. If a sequence$x=(x_j)$is generalized statistically τ-convergent to a number ξ, then it is$S_{\lambda }^{\ast }(\tau )$-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

$$K_i=\{j\in \mathbb{N}:x_j-\xi \notin U_i\}.$$

By hypothesis, ${\delta }_{\lambda }(K_i)=0$ for each i. Since the ideal ℐ of all subsets of ℕ having λ-density zero is a P-ideal (see, for instance, [43]), then there exists a sequence of sets ${(J_i)}_i$ 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 ${\delta }_{\lambda }(K)=1$. 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

$$(\mathbb{N}\setminus J_i)\cap \{j\in \mathbb{N}:j\ge j_i\}=(\mathbb{N}\setminus K_i)\cap \{j\in \mathbb{N}:j\ge j_i\}.$$

(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., [44–49]). 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.