On ideal convergence of double sequences in probabilistic normed spaces (Q1757950)

The authors introduce the notion of ideal convergence for double sequences in a probabilistic normed (=PN) space. Given a PN space \((X,\nu,T)\) and a non-trivial ideal \(\oint\) in \(\mathbb{N}\times\mathbb{N}\), a double sequence \((x_{ij})\) of elements of \(X\) is said to be \(\oint\)--convergent to \(\xi\in X\) if, for all \(\varepsilon>0\) and \(\lambda\in (0,1)\) the set \(\{(i,j)\in\mathbb{N}\times\mathbb{N}:\nu_{x_{ij}-\xi}(\varepsilon)\leq 1-\lambda\}\) belongs to \(\oint\). The \(\oint\)-limit is unique; moreover, the authors prove that, for suitable choices of the ideal, \(\oint\)-convergence implies the usual convergence in the PN space, statistical convergence, logarithmic statistical convergence, uniform statistical convergence, as well as more technical results.