On Spătaru's extension of the Hsu-Robbins-Erdös law of large numbers
The author considers a sequence \((X_n )_{n\in \mathbb{N}}\) of independent random variables whose set of distributions has \(p\) elements. More precisely, suppose that the set \(\mathbb{N}\) is partitioned into \(p\) infinite subsets \(\mathbb{N}_1, \dots, \mathbb{N}_p\), and let \(H_1, \dots, H_p\) be \(p\) distributions. Then assume that the distribution of \(X_n\) is \(H_i\) whenever \(n\in \mathbb{N}_i\), \(1\leq i\leq p\). Under this setting, the reviewer [J. Math. Anal. Appl. 187, No. 2, 371-383 (1994)] proved that (1) \(\sum^\infty_{n=1} P(|S_n |> \varepsilon n)< \infty\), \(\varepsilon> 0\), where \(S_n= X_1+ \dots+ X_n\), always implies (2) \(\sum^\infty_{n=1} \sum^n_{k=1} P(|X_k|\geq n)< \infty\) and (3) \(\lim_{n\to \infty} {1\over n} \sum^n_{k=1} E[X_k 1_{\{|X_k|< n\}} ]=0\), while, conversely, if (2) and (3) hold, and other two auxiliary conditions are fulfilled, then (1) holds. This generalizes the Hsu-Robbins-Erdös theorem stating that in the i.i.d. case, i.e. \(p=1\), (1) holds iff \(EX^2_1< \infty\) and \(EX_1 =0\). The main contribution of the present author improves the reviewer's result by removing one of the auxiliary conditions. Namely, for each \(i=1, \dots, p\), let \(\alpha_i (n)= \sharp \{k\in \mathbb{N}_i: k\leq n\}\), and assume there are \(\theta_i> 0\) and \(C>0\) such that \(\sum_{k\in \mathbb{N}_i \cap[ n,\infty)} k^{-\theta_i}\leq C\alpha_i (n)/ n^{\theta_i}\). Then (2) and (3) imply (1). Moreover, the author shows that solely (2) and (3) are not sufficient for (1) to hold, as the reviewer conjectured.
scientific article; zbMATH DE number 902561