Zero-one laws and weak convergences for sums of independent random variables

✍ Scribed by G. Siegel

• Publisher: John Wiley and Sons
• Year: 1978
• Tongue: English
• Weight: 526 KB
• Volume: 86
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19780860124

✦ Synopsis

Let mlzk be a median of X,, and put S, = X,, + + X , + . . . +Xnkrt-A,, where {A,, n= 1, 2, . . .} is a sequence of constants.

S, and X,, are subject to F, and F,,, respectively. The problem is the existence of a non-defective limit distribution for {Fn, n = 1, 2, . . .} in the sence of weak convergence. The following zero-one theorem is our main result, where d = F ( m ) --P( --a).

Theorem 1.1. (i) If kn C P (IX,-mnkI>x)+O k = i (1.1) uniformly in n and (1.2) F,*F (n--) ,**) then we have either d = 0 or d = 1. (ii) Prom (1.2) and d = 1 si t follows (1.1). Put &(x, F ) = s u p (F ( t + x ) -F ( t ) ) . The sequence {F,, n = l , 2, . . .} is called compact (weakly), if every subsequence contains cl, subsequence that converges compIetely, i.e. F,*F, a(-) = 1 , F( -a) = 0. Then we have Theorem 1.2. A sequence of constants {An, n'-1, 2, . . .} such that {Fn, n = 1 , 2 , . . .> is compact exists if and only if ( I . 1 ) and (1.3) lim inf &(x, P,L) r O n +for some R: > 0. In this case we c c ~n choose for some 7' =-0 *) uberarbeitete Fassung eingereicht am 11. 1. 1978.

**)means weak convergence which is not nec,essarily complete.