ALMOST SURE CONVERGENCE FOR WEIGHTED SUM OF I.I.D. RANDOM VARIABLES (Ⅱ)

다국어 초록 (Multilingual Abstract)

Let {X,Xn,n≥1} be a sequence of independent and identically distributed(i.i.d.) random variables with EX=0 and E|X|^p＜∞ for some p≥1. Let {a_ni,1≤i≤n,n≥1} be a triangular array of constants. The almost sure(a.s.) convergence of weighted sums ∑^n_i=1a_niX_i can be founded in Choi and Sung[1], Chow[2] and Lai[3], Li et al.[4], Stout[6], Sung[8], Teicher[9], and Thrum[10]. As a special case of general statements, Teicher[9, p.341] obtained the following:
Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with EX=0. If max_(1≤i≤n)|a_ni|=O(1/(m^(1/p)logn)) and E|X|^p＜∞(1≤p≤2), then ∑^n_i=1a_niX_i converges to zero a.s.
Choi and Sung[1] and Sung[8](p=1 and 1＜p＜2, respectively) proved Teicher's result under the weaker condition max_(1≤i≤n)|a_ni|=O(1/(m^(1/p)(logn)^(1-1/p))). The purpose of this paper is to weaken Teicher's condition max_(1≤i≤n)|a_ni|=O(1/(m^(1/p)logn)) for the case p=2.
In what follows we will use the following notation: logx=ln max{x, e}, where ln is the natural logarithm, and C denotes a positive constant which is not necessarily the same one in each appearance.