ON ASYMPTOTICALLY LACUNARY STATISTICAL EQUIVALENT SEQUENCES IN PROBALISTIC NORMED SPACE

Marouf (1993) presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. Patterson (2003), extended those concepts by presenting an asymptotically statistica l equivalent analog of these definitions and natural regularity conditions for non-negative sum ability matrices. In Patterson and Savas (2006) extended th  definitions presented in (Patterson, 2003) to lacun ry sequences. This study extends the definitions presented in (Patterson and Savas, 2006) to lacunar y sequences in probabilistic normed space. An interesting and important generalization of the notion of metric space was introduced by Menger (1942) under the name of statistical metric, which is now called probabilistic metric space. The notion o f a probabilistic metric space corresponds to the situations when we do not know exactly the distance  between two points; we know only probabilities of possible values of this distance. The theory of probabilistic metric space was developed by numerou s authors, as it can be realized upon consulting the list of references in (Constantin and Istratescu, 1989), as well as those in (Schweizer and Sklar, 1960; 1983).  Probabilistic normed spaces (briefly, PN-spaces) ar e linear spaces in which the norm of each vector is a n appropriate probability distribution function rathe r than a number. Such spaces were introduced by Serstnev (1963). Alsina et al. (1993), gave a new definition of PN-spaces which includes Serstnev’s a  special case and leads naturally to the identificat on of the principle class of PN-spaces, the Menger spaces . Important families of probabilistic metric spaces a re probabilistic normed spaces. The theory of probabil istic normed spaces is important as a generalization of deterministic results of linear normed space. It seems therefore reasonable to think if the conce pt of statistical convergence can be extended to probabilistic normed spaces and in that case enquir e how the basic properties are affected. But basic proper ties do not hold on probabilistic normed spaces. The proble m is that the triangle functions in such spaces. In this study we study the concept of asymptoticall y lacunary statistical convergent sequences on probab ilistic normed spaces. Since the study of convergence in PN spaces is fundamental to probabilistic functional a nalysis, we feel that the concept of symptotically lacunary statistical convergent sequences in a PN-space woul d provide a more general framework for the subject.


INTRODUCTION
presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. Patterson (2003), extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for non-negative sum ability matrices. In Patterson and Savas (2006) extended the definitions presented in (Patterson, 2003) to lacunary sequences. This study extends the definitions presented in (Patterson and Savas, 2006) to lacunary sequences in probabilistic normed space.
An interesting and important generalization of the notion of metric space was introduced by Menger (1942) under the name of statistical metric, which is now called probabilistic metric space. The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance between two points; we know only probabilities of possible values of this distance. The theory of probabilistic metric space was developed by numerous authors, as it can be realized upon consulting the list of references in (Constantin and Istratescu, 1989), as well as those in (Schweizer and Sklar, 1960;1983). Probabilistic normed spaces (briefly, PN-spaces) are linear spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. Such spaces were introduced by Serstnev (1963). Alsina et al. (1993), gave a new definition of PN-spaces which includes Serstnev's a special case and leads naturally to the identification of the principle class of PN-spaces, the Menger spaces. Important families of probabilistic metric spaces are probabilistic normed spaces. The theory of probabilistic normed spaces is important as a generalization of deterministic results of linear normed space.
It seems therefore reasonable to think if the concept of statistical convergence can be extended to probabilistic normed spaces and in that case enquire how the basic properties are affected. But basic properties do not hold on probabilistic normed spaces. The problem is that the triangle functions in such spaces.
In this study we study the concept of asymptotically lacunary statistical convergent sequences on probabilistic normed spaces. Since the study of convergence in PNspaces is fundamental to probabilistic functional analysis, we feel that the concept of symptotically lacunary statistical convergent sequences in a PN-space would provide a more general framework for the subject.

Prelimiaries
Now we recall some notations and definitions used in study. Marouf (1993) two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically equivalent if: k xk lim = 1, (denoted by x y) yk ∼ Freedmann et al. (1978) let K be a subset of ℕ ; the set of natural numbers. Then the asymptotic density of K, denoted by δ (K), is defined as:

JMSS Definition 2.2
where the vertical bars denote the cardinality of the enclosed set.

Definition 2.3
Fridy (1988) a number sequence x = (x k ) is said to be statistically convergent to the number l if for each ε > 0, the set K (ε) = {k ≤ n:x k −l≥ε} has asymptotic density zero, i.e.: In this case we write st−lim x = l. The next definition is natural combination of definitions 2.1 and 2.2.
Definition 2.4 Patterson (2003) two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically statistical equivalent of multiple l provided that for every ε > 0: and simply asymptotically statistical equivalent if l = 1. Let S denote all sequences x = (x k ) and y = (y k ) such that l x S y ɶ . By a lacunary sequence θ = (k r ) r = 0, 1, 2,.., where k o = 0; we shall mean an increasing sequence of nonnegative integers with h r = k r −k r −1→∞ as r→∞. The intervals determined by θ will be denoted by I r = (k r−1 , k r ] and the ratio r r 1 k k − will be denoted by q r . Definition 2.5 Patterson and Savas (2006) let θ = (k r ) be a lacunary sequence, the two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically lacunary statistical equivalent of multiple l provided that ε>0: and simply asymptotically lacunary statistical equivalent if l = 1. Let S θ denote all sequences x = (x k ) and y = (y k ) such that l x S y θ ɶ . For the following concepts, we refer to Menger (1942) and Schweizer and Sklar (1960;1983).

Example 2.1
The > operations a > b = max {a + b − 1, 0}, a > b = a.b and a > b = min {a, b} on [0, 1] are t-norms. Schweizer and Sklar (1960;1983) a triple (X, N, >) is called a probabilistic normed space or shortly PN-space if X is a real vector space, N is a mapping from X into D (for x∈X, the distribution function N(x) is denoted by N x and N x (t) is the value of N x at t ∈ ℝ ) and > is a t-norm satisfying the following conditions:

Example 2.2
Suppose that (X, ||.||) is a normed space µ ∈ D with µ(0) = 0 and µ ≠ h, where: where, x ∈ X, t ∈ ℝ . Then (X, N, >) is a PN-space. For example if we define the functions µ and ν on ℝ by: Then we obtain the following well-known > norms: We recall the concepts of convergence and Cauchy sequences in a probabilistic normed space.

Definition 2.9
Asadollah and Nourouzi (2008) let (X, N, >) is a PNspace. Then a sequence x = (x k ) is said to be convergent to l∈X with respect to the probabilistic norm N if, for every ε > 0 and θ ∈ (0, 1), there exists a positive integer k o such that N xk−l (ε) > 1 − θ whenever k ≥ k o . It is denoted by N − lim x = L or N k x L  → as k→∞.

Definition 2.10
Asadollah and Nourouzi (2008) let (X, N, >) is a PNspace. Then a sequence x = (x k ) is called a Cauchy sequence with respect to the probabilistic norm N if, for every ε>0 and θ∈(0, 1), there exists a positive integer k o such that N xk−xl (ε) > 1−θ for all k, l ≥ k o . Asadollah and Nourouzi (2008) let (X, N, >) is a PNspace. Then a sequence x = (x k ) is said to be bounded in X, if there is a r ∈ ℝ such that N xk (r) > 1−θ, where θ ∈ (0, 1). We denote by N l ∞ the space of all bounded sequences in PN space.

Asymptotically Lacunary Convergence on PN-Spaces
The idea of statistical convergence was first introduced by Steinhaus (1951) and then studied by various authors, e.g., (Salat, 1980;Fridy, 1988;Connor, 1988;Esi, 1996) and many others and in normed space by Kolk (1988). Recently Karakus (2007) has studied the concept of statistical convergence in probabilistic normed spaces.
Definition 3.1 Karakus (2007) let (X, N, >) is a PN-space. Then a sequence x = (x k ) is said to be statistically convergent to l ∈ X with respect to the probabilistic norm N provided that for every ε > 0 and γ ∈ (0, 1): In this case we write st N − lim x = l. We are now ready to obtain our main results.

Definition 3.2
Let (X, N, >) is a PN-space and θ = (k r ) be a lacunary sequence. The two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically lacunary statistical equivalent of multiple l in PN-space X if for every ε > 0 and γ∈ (0, 1) Equation 1: r k l yk Or equivalently: r r r k l yk In this case we write x l S (PN) θ ɶ y and simply asymptotically lacunary statistical equivalent if l = 1. Furthermore, let l S (PN) θ ɶ denote the set of all sequences x = (x k ) and y = (y k ) such that x l S (PN) θ ɶ y. By using (3.1) and well-known density properties, we easily get the following lemma.

Theorem 3.2
Let (X, N, >) is a PN-space. If two sequences x = (x k ) and y = (y k ) are asymptotically lacunary statistical equivalent of multiple l with respect to the probabilistic norm N, then l is unique.