Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology

Abstract

In this paper, the concept of probabilistic g-metric space with degree l, which is a generalization of probabilistic G-metric space, is introduced. Then, by endowing strong topology, the definition of l-dimensional asymptotic density of a subset A of $\mathbb{N}^{l}$ is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.

1 Introduction

The theory of probabilistic metric space (PM-space) as a generalization of ordinary metric space was introduced by Menger in [12]. In this space, distribution functions are considered as the distance of a pair of points in statistics rather than deterministic.

The concept of the generalized metric space (briefly G-metric space) was introduced by Mustafa and Sims in 2006 [16]. Then, in 2014, Zhou et al. [26] generalized the notion of PM-space to the G-metric spaces and defined the probabilistic generalized metric space which is denoted by PGM-space.

In [3], Choi et al. proposed a generalization of G-metric space named g-metric space with degree l, in which the distance function with degrees $l=1,2$ is equivalent to ordinary and G-metric, respectively.

The idea of statistical convergence was first introduced by Steinhaus [25] for real sequences and developed by Fast [7], then reintroduced by Shoenberg [22]. Many authors, such as [4, 6, 8, 9, 17, 21], have discussed and developed this concept. The theory of statistical convergence has many applications in various fields such as approximation theory [5], finitely additive set functions [4], trigonometric series [27], and locally convex spaces [11].

In 2008, Sencimen and Pehlivan [24] introduced the concepts of statistically convergent sequence and statistically Cauchy sequence in the probabilistic metric space endowed with strong topology.

The purpose of this paper is to develop a concept to generalize the probabilistic G-metric space to the probabilistic g-metric space with degree l. Here, the notation of the generalized space is still referred as PGM-space. The l-dimensional asymptotic density of a subset A of $\mathbb{N}^{l}$ defined previously by the author in [1] is used to introduce the statistically convergent and Cauchy sequences with respect to strong topology, and some basic facts are studied. Note that in this definition $l=1$ and $l=2$ values coincide exactly with the statistical convergence in PM-space and PGM-space (related to G-metric), respectively. Thus, the definitions and the obtained results show that this study is more comprehensive.

2 Preliminaries

In this section, some basic definitions and results related to PM-space, PGM-space, and statistical convergence are presented and discussed. First, recall the definition of triangular norm (t-norm) as follows.

Definition 2.1

([23])

A mapping $T:[0,1]\times [0,1] \to [0,1]$ is called a continuous t-norm if T satisfies the following conditions:

• (i) T is commutative and associative, i.e., $T(a,b)=T(b,a)$ and $T(a,T(b,c))=T(T(a,b),c)$ for all $a,b,c\in [0,1]$;

• (ii) T is continuous;

• (iii) $T(a,1)=a$ for all $a\in [0,1]$;

• (iv) $T(a,b)\leq T(c,d)$ whenever $a\leq c$ and $b\leq d$ for all $a,b,c,d\in [0,1]$.

A distribution function F is a map from extended reals $\mathbb{R}_{\infty }:=\mathbb{R}\cup \{-\infty , \infty \}$ into $[0, 1]$ such that it is nondecreasing, left-continuous at every real number, and $F(-\infty )=0$ and $F(\infty )=1$. The set of all distribution functions is denoted by Δ and $\Delta ^{+}:=\{F\in \Delta : F(0)=0\}$.

Definition 2.2

([23])

A Menger probabilistic metric space (PM-space) is a triple $(X,F,T)$, where X is a nonempty set, T is a continuous t-norm. and F is a mapping from $X\times X\to \Delta ^{+}$ satisfying the following conditions:

($F_{(x,y)}$ denotes the value of F at the pair $(x,y)$)

• (i) $F_{(x,y)}(t)=1$ for all $x,y\in X$ and $t>0$ if and only if $x=y$;

• (ii) $F_{(x,y)}(t)=F_{(y,x)}(t)$;

• (iii) $F_{(x,y)}(t+s)\geq T (F_{(x,z)}(t), F_{(z,y)}(s) )$ for all $x,y,z\in X$ and $t,s\geq 0$.

Definition 2.3

([16])

Let X be a nonempty set and $G:X\times X \times X \to \mathbb{R}^{+}$, be a function satisfying:

• 1) $G(x,y,z)=0$ if $x=y=z$;

• 2) $0< G(x,x,y)$ for all $x,y \in X$ with $x\neq y$;

• 3) $G(x,x,y)\leq G(x,y,z)$ for all $x,y,z\in X$ with $z\neq y$;

• 4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots$ (symmetry in all three variables);

• 5) $G(x,y,z)\leq G(x,a,a)+G(a,y,z)$ for all $x,y,z,a \in X$.

Then the pair $(X, G)$ is called G-metric space.

The following definition is a developing of PM-space on G-metric.

Definition 2.4

([26])

A Menger probabilistic G-metric space (PGM-space) is a triple $(X, G^{*}, T)$, where X is a nonempty set, T is a continuous t-norm, and $G^{*}$ is a mapping from $X\times X \times X$ into $\Delta ^{+}$, satisfying the following conditions:

• (i) $G^{*}_{(x,y,z)}(t)=1$ for all $x, y,z \in X$ and $t>0$ if and only if $x=y=z$;

• (ii) $G^{*}_{(x,x,y)}(t)\geq G^{*}_{(x,y,z)}(t)$ for all $x,y\in X$ with $z\neq y$ and $t>0$;

• (iii) $G^{*}_{(x,y,z)}(t)=G^{*}_{(x,z,y)}(t)=G^{*}_{(y,x,z)}(t)=\cdots$ (symmetry in all three variables);

• (iv) $G^{*}_{(x,y,z)}(t+s)\geq T(G^{*}_{(x,a,a)}(t), G^{*}_{(a,y,z)}(s))$ for all $x,y,z,a\in X$ and $s,t\geq 0$.

Definition 2.5

([26])

Let $(X, G^{*}, T)$ be a PGM-space and $x_{0}\in X$. For $\epsilon >0$ and $0<\delta <1$, the $(\epsilon , \delta )$-neighborhood of $x_{0}$ is defined as follows:

$$ N_{x_{0}}(\epsilon , \delta )= \bigl\{ y\in X: G^{*}_{(x_{0}, y,y)}( \epsilon )>1-\delta , G^{*}_{(y, x_{0},x_{0})}(\epsilon )>1-\delta \bigr\} . $$

Definition 2.6

([26])

• (i) A sequence $\{x_{n}\}$ in a PGM-space $(X, G^{*}, T)$ is said to be convergent to a point $x\in X$ if, for every $\epsilon >0$ and $0<\delta <1$, there exists a positive integer $M_{\epsilon ,\delta }$ such that $x_{n}\in N_{x}(\epsilon , \delta )$ whenever $n>M_{\epsilon ,\delta }$.

• (ii) A sequence $\{x_{n}\}$ in a PGM-space $(X, G^{*}, T)$ is called a Cauchy sequence if, for every $\epsilon >0$ and $0<\delta <1$, there exists a positive integer $M_{\epsilon ,\delta }$ such that $G^{*}_{(x_{m}, x_{n},x_{l})}(\epsilon )>1-\delta$ whenever $m, n, l>M_{\epsilon ,\delta }$.

• (iii) A PGM-space $(X, G^{*}, T)$ is said to be complete if every Cauchy sequence in X converges to a point in X.

In the following, some basic concepts of statistical convergence are discussed.

Definition 2.7

([7])

Let $A\subset \mathbb{N}$ and $A(n)=\{k\in A ; k\leq n\}$. Then the asymptotic density of A is defined as follows:

$$ \delta (A)=\lim_{n\to \infty }\frac{ \vert A(n) \vert }{n}. $$

For a subset A of $\mathbb{N}$, if $\delta (A)=1$, then it is said to be statistically dense. It is clear that $\delta (\mathbb{N}-A)=1-\delta (A)$.

Definition 2.8

([7])

A sequence $\{x_{n}\}$ in $\mathbb{R}$ is said to be statistically convergent to a point x in $\mathbb{R}$ if, for each $\epsilon >0$,

$$ \lim_{n\to \infty }\frac{1}{n} \bigl\vert \bigl\{ k\leq n: \vert x_{k}-x \vert \geq \epsilon \bigr\} \bigr\vert =0. $$

For more information about statistical convergence, the references [2, 4, 7–10, 13–15, 18–20] can be addressed.

3 Main results

In this section the main definitions and results are introduced and discussed. First of all, consider the following definition which is a generalization of a G-metric space to an l-dimensional case, where $l\in \mathbb{N}$.

Definition 3.1

([3])

Let X be a nonempty set. A function $g:X^{l+1}\longrightarrow \mathbb{R}_{+}$ is called a g-metric with degree l on X if it satisfies the following conditions:

• g1) $g(x_{0}, x_{1},\ldots, x_{l})=0$ if and only if $x_{0}=x_{1}=\cdots=x_{l}$,

• g2) $g(x_{0}, x_{1},\ldots, x_{l})=g(x_{\sigma (0)}, x_{\sigma (1)},\ldots,\ldots,x_{ \sigma (l)})$ for permutation σ on $\{0, 1,\ldots, l\}$,

• g3) $g(x_{0}, x_{1},\ldots, x_{l})\leq g(y_{0}, y_{1},\ldots, y_{l})$ for all $(x_{0}, x_{1},\ldots, x_{l}), (y_{0}, y_{1},\ldots, y_{l})\in X^{l+1}$ with $\{x_{i}: i=0, 1,\ldots, l\}\subseteq \{y_{i}: i=0, 1,\ldots, l\}$,

• g4) For all $x_{0}, x_{1},\ldots, x_{s}, y_{0}, y_{1},\ldots, y_{t}, w\in X$ with $s+t+1=l$,

  $$ g(x_{0}, x_{1},\ldots, x_{s}, y_{0}, y_{1},\ldots, y_{t})\leq g(x_{0}, x_{1},\ldots, x_{s}, w, w,\ldots, w)+g(y_{0}, y_{1},\ldots, y_{t}, w, w,\ldots, w). $$

The pair $(X,g)$ is called a g-metric space. It is noteworthy that, if $l=1$ (resp. $l=2$), then it is equivalent to an ordinary metric space (resp. G-metric space).

Definition 3.2

([3])

Let $(X,g)$ be a g-metric space, $x\in X$ be a point, and $\{x_{k}\}\subseteq X$ be a sequence.

• 1) $\{x_{k}\}$ g-converges to x if for all $\epsilon >0$ there exists $N\in \mathbb{N}$ such that

  $$ i_{1},\ldots, i_{l}\geq N\quad \Longrightarrow \quad g(x, x_{1},\ldots, x_{l})< \epsilon . $$

• 2) $\{x_{k}\}$ is said to be g-Cauchy if for all $\epsilon >0$ there exists $N\in \mathbb{N}$ such that

  $$ i_{0}, i_{1},\ldots, i_{l}\geq N\quad \Longrightarrow\quad g(x_{i_{0}}, x_{i_{1}},\ldots, x_{i_{l}})< \epsilon . $$

• 3) $(X, g)$ is complete if every g-Cauchy sequence in $(X, g)$ is g-convergent in $(X, g)$.

Now, by equipping Definition 2.4 with g-metric, we introduce the following definition that is a generalization.

Definition 3.3

A Menger probabilistic g-metric space (still is denoted as PGM-space) is a triple $(X, F, T)$, where X is a nonempty set, T is a continuous t-norm, and F is a mapping from $X^{l+1}$ into $\Delta ^{+}$, satisfying the following conditions:

• (i) $F_{(x_{0}, x_{1},\ldots, x_{l})}(t)=1$ for all $x_{0}, x_{1},\ldots, x_{l} \in X$ and $t>0$ if and only if $x_{0}= x_{1}= \cdots= x_{l}$;

• (ii) $F_{(x_{0}, x_{1},\ldots, x_{l})}(t)\geq F_{(y_{0}, y_{1},\ldots, y_{l})}(t)$ for all $(x_{0}, x_{1},\ldots, x_{l}), (y_{0}, y_{1},\ldots, y_{l})\in X^{l+1}$ with $\{x_{i}: i=0, 1,\ldots, l\}\subseteq \{y_{i}: i=0, 1,\ldots, l\}$;

• (iii) $F_{(x_{0}, x_{1},\ldots, x_{l})}(t)=F_{(x_{\sigma (0)}, x_{\sigma (1)},\ldots,\ldots,x_{\sigma (l)})}(t)$ for permutation σ on $\{0, 1,\ldots, l\}$;

• (iv) For all $x_{0}, x_{1},\ldots, x_{m}, y_{0}, y_{1},\ldots, y_{n}, w\in X$ with $m+n+1=l$,

  $$ F_{(x_{0}, x_{1},\ldots, x_{m}, y_{0}, y_{1},\ldots, y_{n})}(t+s)\geq T\bigl(F_{(x_{0}, x_{1},\ldots, x_{m}, w, w,\ldots, w)}(t), F_{(y_{0}, y_{1},\ldots, y_{n}, w, w,\ldots, w)}(s)\bigr). $$

In the following, according to the generalization of asymptotic density given in [1], statistically convergent and Cauchy sequences in a PGM-space are introduced.

Definition 3.4

Let $(X, F,T)$ be a PGM-space. For any $\epsilon >0$, $0<\delta <1$ and $x\in X$, the strong $(\epsilon , \delta )$-vicinity of x is defined by the subset $M_{x}(\epsilon , \delta )$ of $X^{l}$ as follows:

$$ M_{x}(\epsilon , \delta )=\bigl\{ (x_{1}, x_{2}, \ldots, x_{l})\in X^{l}; F_{(x, x_{1}, x_{2},\ldots, x_{l})}(\epsilon )>1-\delta \bigr\} . $$

Next, we generalize the concept of asymptotic density of a set in an l-dimensional case.

Definition 3.5

Let $K\subset \mathbb{N}^{l}$, the l-dimensional asymptotic density of K is defined by

$$ \delta _{l}(K)=\lim_{n\to \infty }\frac{l!}{n^{l}} \bigl\vert \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in K; i_{1}, i_{2},\ldots, i_{l} \leq n\bigr\} \bigr\vert . $$

Definition 3.6

Let $(X, F,T)$ be a PGM-space.

• (i) A sequence $\{x_{n}\}$ in X is statistically convergent to a point x in X w.r.t. strong topology if, for any $\epsilon >0$ and $0<\delta <1$,

  $$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )\leq 1- \delta \bigr\} \bigr)=0, $$

  and is denoted by $x_{n} \overset{st}{\longrightarrow } x$ or $st-\lim_{n\to \infty }x_{n}=x$.

• (ii) $\{x_{n}\}$ is said to be statistically Cauchy w.r.t. strong topology if, for all $\epsilon >0$ and $0<\delta <1$, there exists $i_{\epsilon }\in \mathbb{N}$ such that

  $$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}(\epsilon )\leq 1- \delta \bigr\} \bigr)=0. $$

We can restate part $(i)$ of the above definition as follows:

($i^{\prime }$):

$x_{n} \overset{st}{\longrightarrow } x$ if and only if, for any $\epsilon >0$ and $0<\delta <1$,

$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: (x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}})\notin M_{x}(\epsilon , \delta )\bigr\} \bigr)=0. $$

The following theorem shows that if a sequence is statistically convergent to a point in X, then that point is unique.

Theorem 3.7

Let $\{x_{n}\}$ be a sequence in a PGM-space $(X, F, T)$ such that $x_{n} \overset{st}{\longrightarrow } x$ and $x_{n} \overset{st}{\longrightarrow } y$, then $x=y$.

Proof

Let $\epsilon >0$ and $0<\delta <1$, by the continuity of T, there exists $0<\delta _{0}<1$ such that

$$ T (1-\delta _{0}, 1-\delta _{0} )>1-\delta . $$

Set

$$\begin{aligned}& A(\epsilon ,\delta ):=\biggl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}\biggl( \frac{\epsilon }{2}\biggr)\leq 1-\delta _{0} \biggr\} , \\& B(\epsilon ,\delta ):=\biggl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, y)}\biggl( \frac{\epsilon }{2}\biggr)\leq 1-\delta _{0} \biggr\} , \end{aligned}$$

and

$$ C(\epsilon ,\delta ):=A(\epsilon ,\delta )\cup B(\epsilon ,\delta ). $$

Since $x_{n} \overset{st}{\longrightarrow } x$ and $x_{n} \overset{st}{\longrightarrow } y$, so $\delta _{l}(A(\epsilon ,\delta ))=\delta _{l}(B(\epsilon ,\delta ))=0$ and hence $\delta _{l}(C(\epsilon ,\delta ))=0$, therefore $\delta _{l}(C^{c}(\epsilon ,\delta ))=1$. Suppose $(i_{1}, i_{2},\ldots, i_{l})\in C^{c}(\epsilon ,\delta )$, then by parts (ii) of Definition 3.3 and (iv) of Definition 2.1 we have

$$\begin{aligned} F_{(x, y, y,\ldots, y)}(\epsilon )&\geq T \biggl(F_{(x_{i_{1}}, x_{i_{1}},\ldots, x_{i_{1}}, x)}\biggl( \frac{\epsilon }{2}\biggr), F_{(x_{i_{1}}, y, y,\ldots, y)}\biggl( \frac{\epsilon }{2}\biggr) \biggr) \\ &\geq T \biggl(F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}\biggl( \frac{\epsilon }{2}\biggr), F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, y)}\biggl( \frac{\epsilon }{2}\biggr) \biggr) \\ &> T (1-\delta _{0}, 1-\delta _{0} ) \\ &>1-\delta . \end{aligned}$$

Since $\delta >0$ is arbitrary, we conclude that $F_{(x, y, y,\ldots, y)}(\epsilon )=1$, and therefore $x=y$. □

Theorem 3.8

Every convergent sequence in a PGM-space is statistically convergent.

Proof

Let $\{x_{n}\}$ be a sequence in the PGM-space $(X,F, T)$ that converges to a point $x\in X$. For $\epsilon >0$ and $0<\delta <1$, there exists $n_{0}\in \mathbb{N}$ such that, for all $i_{1}, i_{2},\ldots, i_{l}\geq n_{0}$,

$$ F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )>1-\delta . $$

Set

$$ A(n):=\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: i_{1}, i_{2},\ldots, i_{l}\leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )>1- \delta \bigr\} , $$

then

$$ \bigl\vert A(n) \bigr\vert \geq \binom{n-n_{0}}{l} $$

and

$$ \lim_{n\to \infty }\frac{l! \vert A(n) \vert }{n^{l}}\geq \lim_{n\to \infty } \frac{l!}{n^{l}} \binom{n-n_{0}}{l}=1, $$

so

$$ st-\lim_{n\to \infty } x_{n}=x. $$

 □

Example 3.9 shows that the converse of the above theorem is not valid.

Example 3.9

Let $X=\mathbb{R}$ and $G:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}^{+}$ be a G-metric on $\mathbb{R}$ defined by

$$ G(x,y,z)= \vert x-y \vert + \vert x-z \vert + \vert y-z \vert . $$

$(T= \min )$ Define a function $F:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}^{+}$ as follows:

$$ F_{(x,y,z)}(t)= \textstyle\begin{cases} H(t), & x=y=z, \\ \mathcal{D}{(\frac{t}{G(x,y,z)})}, & \text{otherwise,} \end{cases} $$

where $H(t)$ and $\mathcal{D}(t)$ are distribution functions as follows:

$$ H(t)= \textstyle\begin{cases} 0, & t\leq 0, \\ t, & t>0. \end{cases}\displaystyle ,\quad\quad \mathcal{D}= \textstyle\begin{cases} 0, & t\leq 0, \\ 1-e^{-t}, & t>0. \end{cases} $$

Now, consider the following sequence in $\mathbb{R}$:

$$ x_{n}= \textstyle\begin{cases} n , & n \text{ is square,} \\ 1 , & \text{otherwise.} \end{cases} $$

It is clear that $\{x_{n}\}$ statistically converges to 1 but it is not convergent normally.

Definition 3.10

A set $A=\{n_{k}: k\in \mathbb{N}\}$ is said to be statistically dense in $\mathbb{N}$ if the set

$$ A(n)=\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in A^{l}, i_{1}, i_{2},\ldots, i_{l} \leq n\bigr\} $$

has asymptotic density 1, i.e.,

$$ \delta _{l}(A)=\lim_{n\to \infty }\frac{l! \vert A(n) \vert }{n^{l}}=1. $$

Theorem 3.11

Let $\{x_{n}\}$ be a sequence in the PGM-space $(X, F, T)$. Then the following are equivalent:

• (i) $\{x_{n}\}$ statistically converges to a point $x \in X$.

• (ii) There is a sequence $\{y_{n}\}$ in X such that $x_{n}=y_{n}$ for almost all n, and $\{y_{n}\}$ converges to x.

• (iii) There is a statistically dense subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $\{x_{n_{k}}\}$ is convergent.

• (iv) There is a statistically dense subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $\{x_{n_{k}}\}$ is statistically convergent.

Proof

$(i\Longrightarrow \mathit{ii})$ Let $\{x_{n}\}$ be a sequence that converges to x, so

$$\begin{aligned} &\delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1- \delta \bigr\} \bigr) \\ &\quad =\lim_{n\to \infty }\frac{l!}{n^{l}}\bigl| \bigl\{ (i_{1}, i_{2},\ldots, i_{l}) \in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1-\delta \bigr\} \bigr|=1. \end{aligned}$$

For each $k\in \mathbb{N}$, we can choose an increasing sequence $\{n_{k}\}$ such that, for every $n>n_{k}$,

$$ \frac{l!}{n^{l}} \biggl\vert \biggl\{ (i_{1}, i_{2}, \ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}( \epsilon )> 1- \frac{1}{2^{k}}\biggr\} \biggr\vert >1-\frac{1}{2^{k}}. $$

Define the sequence $\{y_{n}\}$ as follows:

$$ y_{m}= \textstyle\begin{cases} x_{m}, & 1\leq m\leq n_{1}, \\ x_{m}, & n_{k}< m\leq n_{k+1}, i_{1}, i_{2},\ldots, i_{l-1}\leq n_{k+1}, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l-1}}, x_{m}, x)}( \epsilon )> 1-\frac{1}{2^{k}}, \\ x, & \text{otherwise}. \end{cases} $$

Choose $k\in \mathbb{N}$ such that $\frac{1}{2^{k}}<\delta$. It is clear that $\{y_{m}\}$ converges to x. Fix $n\in \mathbb{N}$, for $n_{k}< n\leq n_{k+1}$, we have

$$\begin{aligned} \begin{aligned} &\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n ; x_{i_{j}}\neq y_{i_{j}} \bigr\} \\ &\quad \subseteq \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2}, \ldots, i_{l} \leq n \bigr\} \\ &\quad\quad{} - \biggl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2}, \ldots, i_{l} \leq n_{k} , F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}( \epsilon )> 1- \frac{1}{2^{k}} \biggr\} . \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} &\lim_{n\to \infty }\frac{l!}{n^{l}} \bigl\vert \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n ; x_{i_{j}}\neq y_{i_{j}} \bigr\} \bigr\vert \\ &\quad \leq 1-\lim_{n\to \infty }\frac{l!}{n^{l}} \biggl\vert \biggl\{ (i_{1}, i_{2},\ldots, i_{l}) \in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n_{k} , F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1- \frac{1}{2^{k}} \biggr\} \biggr\vert \\ &\quad < \frac{1}{2^{k}}< \delta , \end{aligned}$$

so

$$\begin{aligned} &\delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2}, \ldots, i_{l} \leq n ; x_{i_{j}}\neq y_{i_{j}} \bigr\} \bigr) \\ &\quad =\lim_{n\to \infty }\frac{l!}{n^{l}} \bigl\vert \bigl\{ (i_{1}, i_{2},\ldots, i_{l}) \in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n ; x_{i_{j}} \neq y_{i_{j}} \bigr\} \bigr\vert =0. \end{aligned}$$

$(\mathit{ii} \Longrightarrow \mathit{iii})$ Let $\{y_{n}\}$ be a convergent sequence in X and $A=\{n\in \mathbb{N}: y_{n}\neq x_{n}\}$. We have $\delta _{l}(A)=1$, so the sequence $\{y_{n}\}$ is a statistical dense subsequence of $\{x_{n}\}$ that is convergent.

$(\mathit{iii} \Longrightarrow \mathit{iv})$ It is obvious from Theorem 3.8.

$(\mathit{iv} \Longrightarrow i)$ Let $\{x_{n_{k}}\}$ be a statistically dense subsequence of $\{x_{n}\}$ that is statistically convergent to a point $x\in X$. Set $A=\{n_{k}: k\in \mathbb{N}\}$, so $\delta _{l}(A)=1$. For ϵ> and $0<\delta <1$,

$$\begin{aligned} &\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1- \delta \bigr\} \\ &\quad \supseteq \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{A}^{l}; i_{1}, i_{2}, \ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}( \epsilon )> 1-\delta \bigr\} . \end{aligned}$$

Hence,

$$\begin{aligned} &\lim_{n\to \infty }\frac{l!}{n^{l}} \bigl\vert \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1-\delta \bigr\} \bigr\vert \\ &\quad \geq \lim_{n\to \infty }\frac{l!}{n^{l}} \bigl\vert \bigl\{ (i_{1}, i_{2},\ldots, i_{l}) \in \mathbb{A}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1-\delta \bigr\} \bigr\vert =1. \end{aligned}$$

So,

$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2}, \ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}( \epsilon )> 1-\delta \bigr\} \bigr)=1. $$

Therefore $\{x_{n}\}$ statistically converges to x. □

The following corollary is a direct consequence of the above theorem.

Corollary 3.12

Every statistically convergent sequence in a PGM-space has a convergent subsequence.

Theorem 3.13

Every statistically convergent sequence in a PGM-space is statistically Cauchy.

Proof

Suppose that $\{x_{n}\}$ is a sequence that statistically converges to a point x. Let $\epsilon >0$ and $0<\delta <1$. Since T is continuous, there are $0<\delta _{1}<1$ and $0<\delta _{2}<1$ such that $T(1-\delta _{1}, 1-\delta _{2})>1-\delta$. On the other hand, there exists $i_{\epsilon }$ such that

$$ F_{(}x_{i_{\epsilon }, x, x,\ldots, x)}\biggl(\frac{\epsilon }{2}\biggr)>1-\delta _{1}. $$

Since

$$ F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )\geq T \biggl(F_{(x_{i_{ \epsilon }}, x,\ldots, x)}\biggl( \frac{\epsilon }{2}\biggr), F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}\biggl(\frac{\epsilon }{2}\biggr) \biggr), $$

so

$$\begin{aligned} & \biggl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{ \epsilon }})}\biggl(\frac{\epsilon }{2}\biggr)> 1- \delta _{2} \biggr\} \\ &\quad \subseteq \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2}, \ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}( \epsilon )> 1-\delta \bigr\} . \end{aligned}$$

Hence

$$\begin{aligned} &\lim_{n\to \infty }\frac{l!}{n^{l}} \biggl\vert \biggl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}\biggl(\frac{\epsilon }{2}\biggr)> 1- \delta _{2} \biggr\} \biggr\vert \\ &\quad \leq \lim_{n\to \infty }\frac{l!}{n^{l}} \bigl\vert \bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}; i_{1}, i_{2},\ldots, i_{l} \leq n, F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )> 1-\delta \bigr\} \bigr\vert . \end{aligned}$$

Since $\{x_{n}\}$ is statistically convergent, so the right-hand side of the previous inequality is zero. Therefore it shows that the sequence $\{x_{n}\}$ is statistically Cauchy. □

Definition 3.14

Let $(X, F,T)$ be a PGM-space. If every statistically Cauchy sequence is statistically convergent, then $(X, F,T)$ is said to be statistically complete.

Corollary 3.15

Every statistically complete PGM-space is complete.

Proof

Let $(X,F,T)$ be a statistically complete PGM-space. Suppose that $\{x_{n}\}$ is a Cauchy sequence in $(X,F,T)$, so it is a statistically Cauchy sequence. Since X is statistically complete, so $\{x_{n}\}$ is statistically convergent. By Corollary 3.12, there is a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ that converges to a point $x\in X$. By the continuity of T, for $0<\delta <1$, there exist $0<\delta _{1}, \delta _{2}, \delta _{3}, \delta _{4}<1$ such that

$$ \textstyle\begin{cases} T(1-\delta _{1}, 1-\delta _{2})>1-\delta , \\ T(1-\delta _{3}, 1-\delta _{4})>1-\delta _{1}. \end{cases} $$

Let $\delta _{5}:=\max \{\delta _{2}, \delta _{3}\}$, then we have

$$ T \bigl(T(1-\delta _{5}, 1-\delta _{4}), 1-\delta _{5} \bigr)>1- \delta . $$

For $\epsilon >0$, since $\{x_{n}\}$ is Cauchy, then there exist $N_{1}\in \mathbb{N}$ and $x_{i_{\epsilon }}\in \{x_{n}\}$ such that, for all $i_{1}, i_{2},\ldots, i_{l}\geq N_{1}$,

$$ F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}\biggl( \frac{\epsilon }{4}\biggr)> 1-\delta _{5}, $$

and since $x_{n_{k}}\longrightarrow x$, there exists $N_{2}\geq N_{1}$ such that, for $i_{n_{1}}, i_{n_{2}},\ldots, i_{n_{l}}\geq N_{2}$,

$$ F_{(x_{i_{n_{1}}}, x_{i_{n_{2}}},\ldots, x_{i_{n_{l}}}, x)}\biggl( \frac{\epsilon }{4}\biggr)> 1-\delta _{5}. $$

For $i_{1}, i_{2},\ldots, i_{l}, i_{n_{1}}, i_{n_{2}},\ldots, i_{n_{l}}\geq N_{2}$, we have

$$\begin{aligned} &F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon ) \\ &\quad \geq T \biggl(F_{(x_{i_{\epsilon }}, x , x,\ldots, x)}\biggl( \frac{\epsilon }{2}\biggr), F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}\biggl(\frac{\epsilon }{2}\biggr) \biggr) \\ &\quad \geq T \biggl(T \biggl(F_{(x_{i_{\epsilon }}, x_{i_{n_{1}}}, x_{i_{n_{1}}},\ldots, x_{i_{n_{1}}})}\biggl(\frac{\epsilon }{4}\biggr), F_{(x_{i_{n_{1}}}, x, x,\ldots, x)}\biggl(\frac{\epsilon }{4}\biggr) \biggr), F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}\biggl( \frac{\epsilon }{2}\biggr) \biggr) \\ &\quad \geq T \biggl(T \biggl(F_{(x_{i_{\epsilon }}, x_{i_{n_{1}}}, x_{i_{n_{2}}},\ldots, x_{i_{n_{l}}})}\biggl(\frac{\epsilon }{4}\biggr), F_{(x_{i_{n_{1}}}, x_{i_{n_{1}}}, x_{i_{n_{2}}},\ldots, x_{i_{n_{l}}})}\biggl(\frac{\epsilon }{4}\biggr) \biggr), F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}\biggl( \frac{\epsilon }{4}\biggr) \biggr) \\ &\quad >T \bigl(T(1-\delta _{5}, 1-\delta _{4}), 1-\delta _{5} \bigr) \\ &\quad >1-\delta . \end{aligned}$$

The third inequality arises from part $(\mathit{ii})$ of Definition 3.3 and the nondecreasing property of F. So, $\{x_{n}\}$ is convergent and therefore $(X,F,T)$ is complete. □