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 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.