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On ideal convergence of double sequences in probabilistic normed spaces. (English) Zbl 1240.40032

An ideal in a nonempty set X is a family I of subsets of X such that (a) XI; (b) A,BIABI, and (c) AI and BIBI. If I is an ideal in X, then F(I)={XA:AI} is a filter on X, called the filter associated to the ideal I. A sequence (x k ) in is called I-convergent to ξ if |[k:|x k -L|ϵ}I for every ϵ>0. I-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) (X,ν,*), where * is a triangle function on [0,1] and ν a probabilistic norm on the real vector space X (see [B. Schweizer and A. Sklar, Probabilistic metric spaces. New York-Amsterdam-Oxford: North-Holland (1983; Zbl 0546.60010)]). For an ideal I 2 in ×, a double sequence x=(x k,j ) in the PNS (X,ν,*) is said to be I 2 ν -convergent to ξX if {(j,k)×:ν x j,k -ξ (t)1-ϵ}I 2 for all ϵ,t>0· If the double sequence x=(x k,j ) is ν-convergent to ξX, then it is I 2 ν -convergent to ξ.

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 I 2 *ν -convergence, and in the last part of the paper study I 2 limit and cluster points for double sequences in PNSs.

MSC:
40J05Summability in abstract structures
46S50Functional analysis in probabilistic metric linear spaces
40A35Ideal and statistical convergence