Distribution of a random power series
In this note the author has generalized the result of \textit{R. Ranga Rao} [Ann. Math. Stat. 33, 659--680 (1962; Zbl 0117.28602)] to the two-dimensional case. Let \(\eta_ i\), \(i=1,2,..\). be independent random vectors of the space \(\mathbb R^ 2\) with identical non-degenerate distribution and let the series \[ \psi =\eta_ 1+\eta_ 2t+\eta_ 3t^ 2+...+\eta_ nt^{n-1}+... \tag{*} \] converge, \(t\in (-1,0)\cup (0,1)\). Then for the boundary \(\Gamma\) of any convex set \(P(\sum^{\infty}_{i=1}\eta_ it^{i-1}\in \Gamma)=0\). He has also proved that the distributions of partial sums of the series (*) converge uniformly on all Borel convex sets in \(\mathbb R^ 2\) if and only if for any point \(x\in \mathbb R^ 2-\{0\}\), the probability \(P(\eta_ i=x)<1\).
scientific article; zbMATH DE number 3921612