Global Journal of Mathematical Analysis, 1 (2) (2013) 29-36 c Science Publishing Corporation www.sciencepubco.com/index.php/GJMA λ 3 -Statistical convergence of triple sequences on probabilistic normed space Ayhan Esi Adiyaman University, Department of Mathematics, 02040, Adiyaman, Turkey E-mail: aesi23@hotmail.com Abstract The idea of λ-statistical convergence of single sequences was studied by Alotaibi [39] and double sequences was studied by Savas and Mohiuddine [3] in probabilistic normed spaces. The purpose of this paper is to study statis- tical convergence of triple sequences in probabilistic normed spaces and give some important theorems. Keywords : λ-sequence, probabilistic normed space, statistical convergence, t-norm. 1 Introduction The concept of statistical convergence play a vital role not only in pure mathematics but also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modelling, and motion planning in robotics. The notion of statistical convergence was introduced by Fast [8] and Schoenberg [32] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated by Fridy [9], ˘ Sal´ at [31], C ¸ akalli [5], Maio and Kocinac [19], Miller [21], Maddox [18], Leindler [17], Mursaleen and Alotaibi [24], Mursaleen and Edely [26], Mursaleen and Edely [28], and many others. In the recent years, generalization of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on Stone- ˘ Cech compactification of the natural numbers. Moreover statistical convergence is closely related to the concept of convergence in probability, (see [6]). The notion of statistical convergence depends on the density of subsets of N. A subset of N is said to have density δ (E) if δ (E) = lim n→∞ 1 n n  k=1 χ E (k) exists. A sequence x =(x k ) is said to be statistically convergent to ℓ if for every ε> 0 δ ({k ∈ N : |x k − ℓ|≥ ε})=0. In this case, we write S − lim x = ℓ or x k → ℓ(S) and S denotes the set of all statistically convergent sequences. The probabilistic metric space was introduced by Menger [20] which is an interesting and important general- ization of the notion of a metric space. Karakus [14] studied the concept of statistical convergence in probabilistic normed spaces. The theory of probabilistic normed spaces was initiated and developed in [4], [33], [34], [35], [37] and further it was extended to random/probabilistic 2-normed spaces by Golet ¸ [11] using the concept of 2-norm which is defined by G¨ahler [10], and G¨ urdal and Pehlivan [13] and [12] studied statistical convergence in 2-Banach spaces.