Statistical Summability through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

Two concepts—one of statistical convergence and the other of de la Vallée-Poussin mean—play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces. 1. Introduction, Definitions, and Preliminaries Fast [1] presented the following definition of statistical convergence for sequences of real numbers. Let , the set of natural numbers, and . The natural density of is defined by if the limit exists, where denotes the cardinality of . The sequence is said to be statistically convergent to the number if for every the set has natural density zero; that is, for each , Note that every convergent sequence is statistically convergent to the same limit, but its converse need not be true. In 1985, Fridy [2] has defined the notion of statistically Cauchy sequence and proved that it is equivalent to statistical convergence and since then a large amount of work has appeared. Various extensions, generalizations, variants, and applications have been given by several authors so far, for example, [3–8] and references therein. In the recent past, Mursaleen [9] presented a generalization of statistical convergence by using de la Vallée-Poussin mean which is known -statistical convergence and further studied by ？olak and Bektas [10, 11]. For more details related to this concept we refer to [12–18]. Let be a nondecreasing sequence of positive numbers tending to such that The generalized de la Vallée-Poussin mean is defined by where . A sequence is said to be -summable to a number if In this case is called -limit of . Let be a set of positive integers; then is said to be -density of . In case , -density reduces to the natural density. Also, since , for every . The number sequence is said to be -statistically convergent to the number if, for each , , where ; that is, In this case we write and we denote the set of all statistically convergent sequences by . A distribution function is an element of , where is left-continuous, nondecreasing, and and the subset is the set . Here denotes the left limit of the function at the point . The space is partially ordered by the usual pointwise ordering of functions; that is, if and only if for all . A triangle function is a binary operation on , namely, a function that is associative, commutative nondecreasing and which has as unit; that is, for all , we have(i) ,(ii) ,(iii) whenever ,(iv) . Here is the d.f. defined by We remark