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On a generalization of the Steinhaus problem on the convergence of series with random signs. (Russian) Zbl 0742.60038

Let the variables \(x_ 1,x_ 2,\dots\) take values \(\pm1\) and the measure \(F_ n\) be generated by \(x_ 1,x_ 2,\dots,x_ n\). It is assumed that for \(n\geq 2\) \[ dP_ n/d\nu_ n=(\hbox{ch}(\sigma))^{- (n-1)}\exp(\alpha\sum_{k=1}^{n-1}x_ k x_{k+1}), \qquad \alpha>0, \] where \(\nu_ n\) is a counting measure on \(Z^ n\). The author proves that for a number sequence \(\{a_ n\}\) \[ S=\sum_ k a_ k x_ k<\infty \iff \sum_ k a^ 2_ k<\infty. \] Two other statements are related to the asymptotic of \(P(| S|>s)\) as \(s\to\infty\).

MSC:

60F99 Limit theorems in probability theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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