Ideal Convergence of Random Variables

The aim of this paper is to introduce and study the notion of -convergence of random variables via probabilistic norms. Furthermore, we introduce -convergence in space and establish some interesting results. 1. Introduction Fast [1] and Steinhaus [2] independently introduced the notion of statistical convergence for sequences of real numbers, which is a generalization of the concept of convergence. The concept of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences through the concept of density. Later on, several generalizations and applications of this concept have been presented by various authors (see [3–10] and references therein). Kostyrko et al. [11] presented a generalization of the concept of statistical convergence with the help of ideal of subsets of the set of natural numbers and further studied in [12–16]. Menger [17] presented an interesting and important generalization of the concept of a metric space under the name of statistical metric space by using probability distribution function, which is now called a probabilistic metric space. By using the concept of Menger, ？erstnev [18] introduced the concept of probabilistic normed space (for random normed space, see [19]), which is an important generalization of deterministic results of linear normed spaces. Afterward, Alsina et al. [20] presented a new definition of probabilistic normed space which includes the definition of ？erstnev as a special case. The concept of ideal convergence for single and double sequence of real numbers in probabilistic normed space was introduced and studied by Mursaleen and Mohiuddine [21, 22]. In the recent past, Mursaleen and Alotaibi [23] and Mohiuddine et al. [24] studied the notion of ideal convergence for single and double sequences in random 2-normed spaces, respectively. For more detail and related concept, we refer to [25–33] and references therein. 2. Basic Definitions and Notations The notion of statistical convergence depends on the density (asymptotic or natural) of subsets of . A subset of is said to have natural density if A sequence is said to be statistically convergent [1] to if for every In this case, we write or , and denotes the set of all statistically convergent sequences. An ideal is defined as a hereditary and additive family of subsets of a nonempty arbitrary set ; here, in our study, it suffices to take as a family of subsets of , positive integers; that is, , such that , for each , and each subset of an element of is an element of . A nonempty family of sets is a filter