Abstract
For a sequence of i.i.d. random variables $\{X, X_{n}, n\ge1\}$ and a sequence of positive real numbers $\{a_{n}, n\ge1\}$ with $0< a_{n}/n^{1/p}\uparrow$ for some $0< p<2$ , the Baum-Katz complete convergence theorem is extended to the $\{X, X_{n}, n\ge1\}$ with the general moment condition $\sum^{\infty}_{n=1}n^{r-1}P\{|X|>a_{n}\}<\infty$ , where $r\ge1$ . The relationship between the complete convergence and the strong law of large numbers is established.
Highlights
Introduction and main resultThe concept of complete convergence was first introduced by Hsu and Robbins [ ] and has played a very important role in probability theory
To prove ( . ), it suffices to prove that a– n S n → in probability
We prove that ( . ) implies ( . )
Summary
The concept of complete convergence was first introduced by Hsu and Robbins [ ] and has played a very important role in probability theory. Hsu and Robbins [ ] proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Their result has been generalized and extended by many authors. It is worth pointing out that Sung [ ] obtained the following complete convergence for pairwise i.i.d. random variables {X, Xn, n ≥ }: n– P n=. Katz [ ] complete convergence and the Marcinkiewicz and Zygmund strong law of large numbers.
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