On the laws of large numbers and moments of supremum of normed sums

Abstract

The purpose of this dissertation is to investigate the moments of maximum of normed sums and the generalizations of SLLN, i.e., the laws of large numbers for Banach valued random variables and the convergence for weighted sums. Let ${Sn, n ≥ 1} denote the partial sums of random variables (Xn). Firstly, the moment conditions for supremum of normed sums in presented when (Xn) are i.i.d. random variables and when (Xn) are martingale differences. When (Xn) are martingale differences, we find a useful sufficient condition of $E(\sup \mid{Sn}\mid^ α/cn) < ∞$, where $0 < cn \mid ∞$ and α is positive constant. From this result, we prove that for $0<p<q< E(\sup \mid{P}/n^p/q) ∞, E(\sup \mid{Xn}\mid{p}/n^p/q< ∞$ and $E\mid{X_1}\mid^q< ∞$ are equivalent for i.i.d. random variables (Xn). Secondly, SLLN for Banach valued random variables (Xn) is studied. We show versions of Chung``s SLLN and Teicher``s SLLN for Banach valued random variables under the assumption that WLLN holds. ◁수식 삽입▷(원문을 참조하세요)