2017/03/24 by Li, Deli, Liang, Han-Ying, Rosalsky, Andrew
#60B12 #60F15 #60G50 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1703.08512
~This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let \X, Xn; n ≥ 1\ be a sequence of independent and identically distributed Banach space valued random variables and set Sn = ∑i=1nXi,~n ≥ 1. Let \an; n ≥ 1\ and \bn; n ≥ 1\ be increasing sequences of positive real numbers such that limn → ∞ an = ∞ and \bn/an;~ n ≥ 1 \ is a nondecreasing sequence. We show that \fracSn- n 𝔼(XI\‖X‖ ≤ bn \ )bn → 0~~almost surely for every Banach space valued random variable X with ∑n=1∞ ℙ(‖X‖ > bn) < ∞ if Sn/an → 0 almost surely for every symmetric Banach space valued random variable X with ∑n=1∞ ℙ(‖X‖ > an) < ∞. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.