Rough Statistical Convergence of Double Sequences in Probabilistic Normed Spaces

In this paper, we have defined rough convergence and rough statistical convergence of double sequences in probabilistic normed spaces which is more generalized version than the rough statistical convergence of double sequences in normed linear spaces. Also, we have defined rough statistical cluster points of double sequences and then, investigated some important results associated with the set of rough statistical limits of double sequences in these spaces. Moreover, in the same spaces, we have proved an important relation between the set of all rough statistical cluster points and rough statistical limits under certain condition.


Introduction
In 1951, the concept of usual convergence of real sequences was extended to statistical convergence of real sequences based on the natural density of a set by Fast [9] and Steinhaus [30] independently.Later on, this idea has been studied in different directions and in different spaces by many authors as in [6,7,8,11,12,20,21,25,32,34] and many more.
In 2001, Phu [27] has initially introduced the concept of rough convergence of sequences in finite dimensional normed linear spaces which is basically a generalization of usual convergence and, in the same paper he has investigated that r-limit set is bounded, closed, convex and many more interesting results and later on, this concept has been extended to infinite dimensional normed linear spaces [29].Also, He [28] has defined the notion of rough continuity of linear operators.Later, Ayter [3] extended this notion to rough statistical convergence based on natural density of a set.Malik and Maity [23,24] has defined rough convergence and rough statistical convergence of double sequences in normed linear spaces.After that, the research work on this concept is still being carried out in different directions as in [4,13,14,18,26] and many references therein.
In 1942, Menger [19] first proposed the concept of statistical metric space, now called probabilistic metric space, which is an interesting and important generalization of the notion of metric space.This concept, later on, was studied by Schweizer and Sklar [33].Combining the idea of statistical metric space and normed linear space, Šerstnev [31] introduced the idea of probabilistic normed space.In 1993 Alsina et al. gave a new definition of probabilistic normed space whic is basically a special case of the definition of Šerstnev.Recently, Antal et al. [5] defined the notion of rough convergence and rough statistical convergence in probabilistic normed spaces.In this space, we have presented the notion of rough statistical convergence of double sequences and investigated some interesting results associated with the sets of rough statistical cluster points and rough statistical limits of double sequences.

Preliminaries
Throughout the paper N and R denote the set of positive integers and set of reals respectively.

Definition 2.2. [10]
A function f : R → R + 0 is said to be a distribution function if it is non decreasing and left continuous with inf t∈R f (t) = 0 and sup t∈R f (t) = 1.We denote D as the set of all distribution functions.Definition 2.3.[10] A triplet (X, ϑ, ⋄) is called a probabilistic normed space (shortly PNS) if X is a real vector space, ν is a mapping from X into D (for x ∈ X, t ∈ (R), ϑ(x; t) is the value of the distribution function ϑ(x) at t) and ⋄ is a t-norm satisfying the following conditions: (1) ϑ(x; 0) = 0; (2) ϑ(x; t) = 1, ∀ t > 0 iff x = θ, θ being the zero element of X; Example 2.2.
Definition 2.7.[5] Let {x n } n∈N be a sequence in an PNS (X, ϑ, ⋄).Then {x n } n∈N is said to be rough convergent to ξ ∈ X with respect to the probabilistic norm ϑ if for every ε > 0, λ ∈ (0, 1) and some non negative number r there exists n 0 ∈ N such that ϑ(x n − ξ; r + ε) > 1 − λ for all n > n 0 .In this case we write r ϑ -lim n→∞ x n = ξ or x n r ϑ − → ξ and ξ is called r ϑ -limit of {x n } n∈N .
Definition 2.8.[5] Let {x n } n∈N be a sequence in an PNS (X, ϑ, ⋆).Then {x n } n∈N is said to be rough statistically convergent to ξ ∈ X with respect to the probabilistic norm ϑ if for every ε > 0 and λ ∈ (0, 1) and some non negative number r, δ({n In this case we write r-St ϑ - Definition 2.9.[21] The double natural density of the set K ⊆ N × N is defined by |{(i, j) ∈ K : i ≤ m and j ≤ n}| mn where |{(i, j) ∈ K : i ≤ m and j ≤ n}| denotes the number of elements of K not exceeding m and n respectively.It can be observed that if K is finite, then δ 2 (K) = 0. Also, if A ⊆ B, then Definition 2.10.[17] Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄).Then {x mn } is said to be statistically convergent to ξ ∈ X with respect to the probabilistic norm ϑ if for every ε > 0 In this case we write

Main Results
First we define rough convergence and rough statistical convergence of double sequences in probabilistic normed spaces.
Definition 3.1.Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄) and r be a non negative real number.Then {x mn } is said to be rough convergent to β ∈ X with respect to the probabilistic norm ϑ if for every ε > 0, λ ∈ (0, 1) there exists Definition 3.2.Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄) and r be a non negative real number.Then {x mn } is said to be rough statistical convergent to β ∈ X with respect to the (b) From Definition 3.1, it is clear that r ϑ 2 -limit of a double sequence may not be unique.So, we denote LIM r ϑ xmn to mean the set of all r ϑ 2 -limit of {x mn } with respect to the probabilistic norm ϑ.
(c) If we put r = 0 in Definition 3.2, then the notion of rough statistical convergence of a double sequence with respect to the probabilistic norm ϑ coincides with statistical convergence of the double sequence with respect to the probabilistic norm ϑ.So, our whole discussion is on the fact r > 0.
(d) From Definition 3.2, it is clear that r-st ϑ 2 -limit of a double sequence may not be unique.So, we denote st ϑ 2 -LIM r xmn to mean the set of all r-st ϑ 2 -limit of {x mn } with respect to the probabilistic norm ϑ.
The sequence {x mn } is said to be r ϑ 2 -convergent if LIM r ϑ xmn = ∅.But, if the sequence is unbounded with respect to the probabilistic norm ϑ then LIM r ϑ xmn = ∅ although in this case st ϑ 2 -LIM r xmn = ∅ may be happened which has been shown in the following example.
Example 3.1.Let (X, • ) be a real normed linear space and let ϑ(x; t) = t t+ x for x ∈ X and t > 0. Then (X, ϑ, ⋄) is a PNS under the t-norm ⋄ defined by x ⋄ y = min{x, y}.For all m, n ∈ N, we define a sequence {x mn } by . Then, Example 3.2.We take the PNS in Example 3.1 and define the double sequence {x mn } by Remark 3.3.From Example 3.2, for any subsequence of a double sequence we not not conclude But, this inclusion may be hold under certain condition which has been given in the following theorem.
Theorem 3.1.Let {x m j n k } be a dense subsequence of {x mn } in a PNS (X, ϑ, ⋄).Then Proof.The proof is obvious.So, we omit details.
Definition 3.3.Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄).Then {x mn } is said to be statistically bounded with respect to the probabilistic norm ϑ if for every λ ∈ (0, 1) there exists Theorem 3.2.Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄).Then {x mn } is statistically bounded if and only if st ϑ 2 -LIM r xmn = ∅ for some r > 0.
Proof.First suppose that {x mn } is statistically bounded.Then for every λ ∈ (0, 1) there exists xmn is closed.This completes the proof.
xmn is convex.This completes the proof.
Theorem 3.6.A double sequence {x mn } in a PNS (X, ϑ, ⋄) is rough statistically convergent to ξ ∈ X with respect to the probabilistic norm ϑ for some r > 0 if there exists a double sequence {y mn } in X such that st ϑ 2 -lim y mn = ξ and for every λ ∈ (0, 1), ϑ(x mn − y mn ; r) > 1 − λ for all m, n ∈ N.
Definition 3.4.(c.f.[16]) Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄).Then a point ξ ∈ X is said to be statistical cluster point of {x mn } with respect to the probabilistic norm ϑ if for every ε > 0 and λ ∈ (0, 1), We denote Λ (xmn) (st ϑ 2 ) to mean ordinary statistical cluster points of {x mn } with respect to the probabilistic norm ϑ.Definition 3.5.Let {x mn } be a double sequence in a PNS (X, ϑ, ⋄).Then a point ξ ∈ X is said to be rough statistical cluster point of {x mn } with respect to the probabilistic norm ϑ if for every ε > 0, λ ∈ (0, 1) and some r > 0, The set of all rough statistical cluster points of {x mn } is denoted as Λ r (xmn) (st ϑ 2 ).

Remark 3 . 1 .
(a) If we put r = 0 in Definition 3.1, then the notion of rough convergence of a double sequence with respect to the probabilistic norm ϑ coincides with notion of ordinary convergence of the double sequence with respect to the probabilistic norm ϑ.