A Refinement of the Kolmogorov-Marcinkiewicz-Zygmund Strong Law of Large Numbers

作者：Li Deli; Qi Yongcheng; Rosalsky Andrew^*

来源：Journal of Theoretical Probability, 2011, 24(4): 1130-1156.

DOI：10.1007/s10959-010-0308-5

• Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers
• Sums of i.i.d. random variables
• Real separable Banach space
• Rademacher type p Banach space
• Stable type p Banach space

Let {X-n; n >= 1} be a sequence of independent copies of a real-valued random variable X and set S-n = X-1 + ... + X-n, n >= 1. This paper is devoted to a refinement of the classical Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. We show that for 0Sigma(infinity)(n=1)1/n(|S-n|n(1/p)) < infinity almost surely
if and only if
{E|X|(p) < infinity if 0 < p < 1,
EX=0, Sigma(infinity)(n=1)|EX1{|X|<= n}|/n < infinity, and
Sigma n=1 infinity integral(n)(min{un,n}) P(|X|>t)dt/n < infinity if p=1,
EX=0 and integral P-infinity(0)1/p(|X|>t)dt < infinity, if 1 < p < 2,
where u(n) = inf{t : P(|X| > t) < 1/n}, n >= 1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jorgensen (Stud. Math. 52: 159-186, 1974) inequality to obtain some general results for sums of the form Sigma(infinity)(n=1) a(n)||Sigma V-n(t=1)i|| (where {V-n; n >= 1} is a sequence of independent Banach-space-valued random variables, and a(n) >= 0, n >= 1), which may be of independent interest, but which we apply to Sigma(infinity)(n=1)1/n(|S-n|n(1/p)).

• 出版日期2011-12