Lacunary statistical convergence of double sequences in topological groups

Recently, Patterson and Savaş (Math. Commun. 10:55-61, 2005), defined the lacunary statistical analog for double sequences x=(x(k,l)) as follows: A real double sequences x=(x(k,l)) is said to be P-lacunary statistically convergent to L provided that for each ε>0, P-limr,s1hr,s|{(k,l)∈Ir,s:|x(k,l)−L|≥ε}|=0. In this case write stθ2-limx=L or x(k,l)→L(stθ2). In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems. MSC:42B15, 40C05.


Introduction
We recall that the concept of statistical convergence of sequences was first introduced by Fast [4] as an extension of the usual concept of sequential limits and also independently Schoenberg [16] for real and complex sequences. Maddox [7] extended statistical convergence to locally convex Hausdorff topological linear spaces in terms of strong summability and further in [6], statistical convergence to normed spaces was extended by Kolk [6]. Also in [1] and [2], Ç akalli extended this notation to topological Hausdorff groups. Savas [15] introduced lacunary statistical convergence of double sequences in topological groups. Also double ideal lacunary statistical convergence in topological groups was studied by Savas (see, [14]). Note that, generalized double statistical convergence in topological groups is considered by Savas, ( see, [13]). More results on double statistical convergence can be seen from [3,11,12].
The notion of the statistical convergence depends on the density of subsets of N A subset E of N is said to have density δ(E) if Note that if K ⊂ N is a finite set, then δ(K) = 0, and for any set K ⊂ N, By X, we will denote an abelian topological Hausdorff group, written additively, which satisfies the first axiom of countability. For a subset B of X, s(B) will denote the set of all sequences x = (x k ) such that x k is in B for k = 1, 2, ..., c(X) will denote the set of all convergent sequences.
A sequence x = (x k ) in X is called to be statistically convergent to an element ξ of X if for each neighborhood U of 0, ( see, [2]) and is called statistically Cauchy in X if for each neighborhood U of 0 there exists a positive integer N = N (ε), depending on the neighborhood U such that lim n→∞ 1 n |{k ≤ n : where the vertical bars indicate the number of elements in the enclosed set. The set of all statistically convergent sequences in X is denoted by S(X) and the set of all statistically Cauchy sequences in X is denoted by SC(X). It is known that By a lacunary sequence, we mean an increasing sequence θ = (k r ) of positive integers such that k 0 = 0 and h r : k r − k r−1 → ∞ as r → ∞. Throughout this paper, the intervals determined by θ will be denoted by I r = (k r−1 , k r ]. Lacunary statistical convergence in topological groups was defined by Cakalli [1] as follows: A sequence x = (x k ) is said to be S θ −convergent to ξ (or lacunary statistically convergent to ξ) if for each neighborhood U of 0, In a topological group X, by the convergence of a double sequence we mean the convergence in Pringsheims sense [10]. A double sequence x = (x kl ) in X is said to be convergent to a point ξ in X in the Pringsheims sense if for every neighborhood U of 0 there exists N ∈ N such that x kl − ξ ∈ U whenever k, l ≥ N . ξ is called the Pringsheim limit of x. A double sequence x = (x kl ) of points in X is said to be a Cauchy sequence if for every neighborhood U of 0 there exists two positive integers N = N (ε) and M = M (ε), depending on the neighborhood U such that The goal of this paper is to introduce the statistical convergence of double sequences in topological groups and to prove some useful theorems.

Definitions and Notation
The double sequence θ = {(k r , l s )} is called double lacunary if there exist two increasing of integers such that kr−1 ,q s = ls ls−1 , and q r,s = q rqs . We will denote the set of all double lacunary sequences by N θr,s . exists.
In 2005, R. F. Patterson and E. Savas [9] studied double lacunary statistically convergence by giving the definition for complex sequences as follows: Definition 1. Let θ be a double lacunary sequence; the double number sequence x is double lacunary statistical convergent to ξ provided that for every ε > 0, In this case write st 2 be a two dimensional set of positive integers and let K m,n be the numbers of (i, j) in K such that i ≤ n and j ≤ m. Then the two-dimensional analogue of natural case density can be defined as follows: The lower asymptotic density of K is defined as In the case when the sequence { Km,n mn } ∞,∞ m,n=1,1 has a limit then we say that K has a natural density and is defined as Recently the studies of double sequences has a rapid growth. The concept of double statistical convergence, for complex case, was introduced by Mursaleen and Edely [8] while the idea of statistical convergence of single sequences was first studied by Fast [4]. Mursaleen and Edely has presented the double statistical convergence as follows: Definition 2. A double sequences x = (x kl ) is said to be P-statistically convergent to ξ provided that for each ε > 0 P − lim m,n 1 mn { number of (k, l) : k < m and l < n, |x kl − ξ| ≥ ε} = 0.
In this case we write st 2 − lim k,l x kl = ξ and we denote the set of all statistical convergent double sequences by st 2 . Recently, statistical convergence of double sequences x = (x kl ) in a topological group was presented by Cakalli and Savas [3] as follows: A sequence x = (x kl ) is called double statistically convergent to a point ξ of X if for each neighborhood U of 0, the set {(k, l), k ≤ n; and; l ≤ m : x kl − ξ / ∈ U } has double natural density zero. In this case we write S 2 (X) − lim k,l x kl = ξ and we write the set of all statistically convergent double sequences by S 2 (X).

Definition 3. (See, [15]).
A sequence x = (x kl ) is said to be S 2 θ −convergent to ξ of X (or lacunary double statistically convergent to ξ of X ) if for each neighborhood U of 0, the set {(k, l) ∈ I rs : x kl − ξ / ∈ U } has double natural density zero. In this case, we write and we write the set of all double lacunary statistically convergent sequences by S 2 θ (X).

Main theorems
Theorem 3.1. A double sequence x = (x kl ) in X is double lacunary statistically convergent to ξ if and only if there exists a subset K ⊂ N × N such that δ 2 θ (K) = 1 and lim k,l→∞ x kl = ξ where limit is being taken over the set X, i.e., (k, l) ∈ X.
Proof. Necessity. Suppose that x = (x kl ) be double lacunary statistically convergent to ξ, and (U i ) be a base of nested closed neighborhoods of 0. Write and Now we shall show that for (k, l) ∈ M i , (x kl ) is double lacunary statistical convergent to ξ. Assume that x = (x kl ) is not double lacunary statistical convergent to ξ so that there is a neighborhood U of 0 such that . Thus x = (x kl ) is convergent to ξ.
Corollary 3.2. If a double sequence x = (x kl ) is double lacunary statistically convergent to ξ, then there exists a double sequence (y kl ) such that lim k,l y kl = ξ and δ 2 θ {(k, l) : x kl = y kl } = 1, i.e., x kl = y kl for almost all (k, l).
In a topological group, double sequence x = (x kl ) is called double lacunary statistically Cauchy if for each neighborhood U of 0 there exists N = N (U ) and M = M (U ) such that the set {(k, l) ∈ I rs : x kl − x N M / ∈ U } has double natural density zero. In this case, we denote the set of all double lacunary statistically Cauchy sequences by S 2 θ C(X). Theorem 3.3. Let X be complete topological group. A double sequence x = (x kl ) in X is double lacunary statistically convergent if and only if x = (x kl ) is double lacunary statistically Cauchy.
We need the following lemma to prove the theorem.
Lemma 3.4. Let be X a topological vector space over the field F . So if W is a neighborhood of 0 in X then there is a neighborhood U of 0 which is symmetric (that is U = −U ) and which satisfies W + W ⊂ U .
Proof. Let x = (x kl ) be double lacunary statistically convergent to ξ. Let U be any neighborhood of 0. Then we may choose a symmetric neighborhood W of 0 such that W + W ⊂ U. Then for this neighborhood W of 0, the set has double natural lacunary density 0. For each neighborhood U of 0, the set {(k, l) ∈ I rs : x kl − ξ / ∈ U } has double natural lacunary density zero. Then we may choose natural numbers M and N such that Therefore we get that x is lacunary statistically Cauchy.
To prove the converse, suppose that there is a double lacunary statistically Cauchy sequence x but it is not double lacunary statistically convergent. Then we may find natural numbers M and N such that the set A U has double natural lacunary density zero. It follows from this that the set has double lacunary natural density 1. Therefore we may choose a symmetric neighborhood W of 0 such that W +W ⊂ U . Now take any fixed non-zero element ξ of X. Let x kl −x M N = x kl −ξ +ξ −x M N . It follows from this equality that x kl −x M N ∈ U if x kl − ξ ∈ W. Since x is not double lacunary statistically convergent to ξ, the set C W has double lacunary natural density 1., i.e., the set {(k, l) ∈ I rs : x kl − ξ / ∈ W } has double lacunary natural density 0. Hence the set {(k, l) ∈ I rs : x kl − x M N ∈ U } has double lacunary natural density 0, i.e., the set A U has double lacunary natural density 1 which is a contradiction. Hence this completes the proof.
Finally we conclude this paper by stating the following theorem which is from theorems 1 and 2 and the proof is easy and omitted.
Theorem 3.5. If X is complete topological group, then the following conditions are equivalent: (a): x is double lacunary statistically convergent to ξ; (b): x is double lacunary statistically Cauchy; (c): there exists a subsequence y of x such that lim k,l y kl = ξ.