Suppose that \(N=\{1,2,\dots\}\) is partitioned into \(p\) infinite subsets \(N_1,\dots,N_p\), and let \(Y_1,\dots,Y_p\) be \(p\) random variables. Let \(\{X_n\}\) be a sequence of independent random variables such that \(X_n\) has the same distribution as \(Y_i\) whenever \(n\in N_i\), \(1\leq i\leq p\). The reviewer [ibid. 187, No. 2, 371-383 (1994; Zbl 0869.60064)] proved that \[ \sum^\infty_{n=1} P(|X_1+\cdots X_n|\geq\varepsilon n) 0,\tag{1} \] always implies \[ (2)\quad \sum^\infty_{n=1} \sum^n_{k-1} P(|X_k|\geq n)<\infty\text{ and}\quad(3)\quad \lim_{n\to\infty} {1\over n}\sum^n_{k=1} E[X_k 1_{\{|X_k|