*(English)*Zbl 1240.40032

An ideal in a nonempty set $X$ is a family $I$ of subsets of $X$ such that (a) $X\in I$; (b) $A,B\in I\Rightarrow A\cup B\in I$, and (c) $A\in I$ and $B\subset I\Rightarrow B\in I$. If $I$ is an ideal in $X$, then $F\left(I\right)=\{X\setminus A:A\in I\}$ is a filter on $X$, called the filter associated to the ideal $I$. A sequence $\left({x}_{k}\right)$ in $\mathbb{R}$ is called $I$-convergent to $\xi \in \mathbb{R}$ if $\left|\right[k\in \mathbb{N}:|{x}_{k}-L|\ge \u03f5\}\in I$ for every $\u03f5>0$. $I$-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) $(X,\nu ,*)$, where $*$ is a triangle function on $[0,1]$ and $\nu $ 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 $\mathbb{N}\times \mathbb{N}$, a double sequence $x=\left({x}_{k,j}\right)$ in the PNS $(X,\nu ,*)$ is said to be ${I}_{2}^{\nu}$-convergent to $\xi \in X$ if $\phantom{\rule{0.277778em}{0ex}}\{(j,k)\in \mathbb{N}\times \mathbb{N}:{\nu}_{{x}_{j,k}-\xi}\left(t\right)\le 1-\u03f5\}\in {I}_{2}$ for all $\u03f5,t>0\xb7$ If the double sequence $x=\left({x}_{k,j}\right)$ is $\nu $-convergent to $\xi \in X$, then it is ${I}_{2}^{\nu}$-convergent to $\xi $.

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 ${I}_{2}^{*\nu}$-convergence, and in the last part of the paper study ${I}_{2}$ limit and cluster points for double sequences in PNSs.