Abstract

For the almost certainly convergent series $S_n=\sum_{i=1}^nV-i$ of independent random elements in Banach spaces, by investigating tail series laws of large numbers, the rate of convergence of the series $S_n$ to a random variable s is studied in this paper. More specifically, by studying the duality between the limiting behavior of the tail series $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}V-i$ of random variables and that of Banach space valued random elements, an alternative way of proving a result of the previous work, which establishes the equivalence between the tail series weak law of large numbers and a limit law, is provided in a Banach space setting.

Keywords

• Rate of Convergence;
• Series of Random Elements in Banach Space;
• Tail Series;
• Strong Law of Large Numbers;
• Weak Law of Large Numbers

• 수렴 속도;
• Banach 공간의 확률요소들의 합;
• Tail 합;
• 강 대수의 법칙;
• 약 대수의 법칙;

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