Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 868457, 8 pages http://dx.doi.org/10.1155/2013/868457 Research Article On Lacunary Ideal Convergence in Random -Normed Space U. YamancJ and M. Gürdal Department of Mathematics, S¨ uleyman Demirel University, East Campus, 32260 Isparta, Turkey Correspondence should be addressed to M. G¨ urdal; gurdalmehmet@sdu.edu.tr Received 27 November 2012; Accepted 4 January 2013 Academic Editor: Baoding Liu Copyright © 2013 U. Yamancı and M. G¨ urdal. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the notions of lacunary I-convergence and lacunary I-Cauchy in the topology induced by random -normed spaces and prove some important results. 1. Introduction Menger [1] generalized the metric axioms by associating a distribution function with each pair of points of a set. Tis system, called a probabilistic metric space, originally a statistical metric space, has been developed extensively by Schweizer and Sklar [2, 3]. Te idea of Menger was to use distribution function instead of nonnegative real numbers as values of the metric, which was further developed by several other authors. In this theory, the notion of distance has a probabilistic nature. Namely, the distance between two points  and  is represented by a distribution function   , and for >0, the value   () is interpreted as the probability that the distance from  to  is less than . Using this concept, ˇ Serstnev [4] introduced the concept of probabilistic normed spaces. It provides an important area into which many deterministic results of linear normed spaces can be generalized. Te studies of continuity properties, linear operators, statistical convergence, and ideal convergence in probabilistic normed spaces have gained many attractions, and such studies have diverse applications into various felds [5–13]. In [14], G¨ ahler introduced an attractive theory of 2- normed and -normed spaces in the 1960s. Since then, many researchers have studied these subjects and obtained various results [15–19]. Since the introduction of the notion of statistical con- vergence for sequences of real numbers by Steinhaus [20] and Fast [21] independently, the theory has been investigated and developed by several authors. Tere has been an efort to introduce several generalizations and variants of statistical convergence in diferent spaces. One very important general- ization of this notion was introduced by Kostyrko et al. [22] by using an ideal I of subsets of the set of natural numbers, which they called I-convergence. Another important variant of statistical convergence is the notion of lacunary statistical convergence introduced by Fridy and Orhan [23]. Recently, Mohiuddine and Aiyub [24] studied lacunary statistical convergence by introducing the concept -statistical convergence in random 2-normed space. Teir work can be considered as a particular generalization of the statistical convergence. In [25], Mursaleen and Mohi- uddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, and Debnath [26] investigated lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Also, lacunary statistically convergent double sequences in probabilistic normed space were studied by Mohiuddine and Savas ¸ in [27]. Te notion of lacunary ideal convergence has not been studied previously in the setting of -normed linear spaces. Motivated by this fact, in this paper, as a variant of I- convergence, the notion of lacunary ideal convergence is introduced in a random -normed space, and some impor- tant results are established. Finally, the notions of lacunary I  -Cauchy and lacunary I  ∗ -Cauchy sequences are intro- duced and studied. Troughout the paper, N will denote the set of all natural numbers. First, we recall some of the basic concepts which will be used in this paper.