An ideal in a nonempty set is a family of subsets of such that (a) ; (b) , and (c) and . If is an ideal in , then is a filter on , called the filter associated to the ideal . A sequence in is called -convergent to if for every . -convergence was considered first by P. Kostyrko, T. Šalát and W. Wilczyński [Real Anal. Exch. 26, No. 2, 669–685 (2001; Zbl 1021.40001)], and independently by F. Nuray and W. H. Ruckle [J. Math. Anal. Appl. 245, No. 2, 513–527 (2000; Zbl 0955.40001)], called by them generalized statistical convergence.
The authors consider this type of convergence for double sequences in a probabilistic normed space (PNS) , where is a triangle function on and a probabilistic norm on the real vector space (see [B. Schweizer and A. Sklar, Probabilistic metric spaces. New York-Amsterdam-Oxford: North-Holland (1983; Zbl 0546.60010)]). For an ideal in , a double sequence in the PNS is said to be -convergent to if for all If the double sequence is -convergent to , then it is -convergent to .
The authors study the basic properties of this type of convergence-characterization in terms of the associated filter, uniqueness, algebraic operations. They consider also a weaker type of convergence, called -convergence, and in the last part of the paper study limit and cluster points for double sequences in PNSs.