Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler. (Analytic continuability of random power series of dependent random variables)
The author considers a sequence \(X=(X_ n(\omega))\) of random variables satisfying the condition \(E(X^ 4_ n)=O(1)\) (n\(\to \infty)\) and studies the corresponding random power series \(f_ x(z,\omega)=\sum^{\infty}_{n=0}a_ nX_ n(\omega)z^ n\). He proves analytic noncontinuability almost surely in the following cases: (i) for \(f_ x\) under the conditions of weak multiplicativity on X; (ii) for \(f_{\tilde X}\) where \(\tilde X=(\tilde X_ n)\) is a subsequence of X; (iii) at least for one of the series \(f_{x'}\) and \(f_{x''}\) where \(X'\) and \(X''\) are certain rearrangements of X defined universally, i.e., independently of the coefficients \(a_ n\).
scientific article; zbMATH DE number 4093823