On Spătaru's Extension of the Hsu-Robbins-Erdős Law of Large Numbers

AbstractThe Hsu-Robbins-Erdo&#x030B;s law of large numbers (1947, 1949) states that ifX1,X2,… are independent identically distributed random variables andSn=X1+···+Xn, then[formula]for every ε>0 if and only ifE[X21]<∞ andE[X1]=0. Under some auxiliary conditions, Spătaru (1994) extended this to the case where theXnare no longer identically distributed, but rather their distributions come from a finite set of distributions. We improve Spătaru's conditions, and present a counterexample to a conjecture of his