Erdös-Rényi law for stationary Gaussian sequences
Yong-Kab Choi · Kyoto journal of mathematics · 1990
IntroductionErdôs a n d Rényi [11] discovered a n e w la w o f l a r g e num bers, now adays called t h e Erd6s-Rényi law.T h is la w s t a t e s t h a t fo r a n i. i. d .sequence fej ; j=1, 2, •-• } w ith partial s u m s S0 = 0 and S = , if th e m o m e n t g e n e ra tin g fu n c tio n M (0 = i=i E exp(te,) e x ists fo r a ll tE(0, t i ), th e n fo r e a c h aE{ } 11/(t)/ill(t); t (0, t1 ) } a n d c=c (a) such that w e have wherem a x O i n -k a n d E-I d en o tes th e integral part.M any general v e r s io n s o f th e ErdOs-Rényi la w f o r i. i. d .sequences h a v e been developed by Book [1]--, [2], NI.Csbrgo [5] , [6], S. Csbrgo [7], Deheuvels [8]-, -, [9] and Steinebach [17] , --[20] a n d others.However, Deo [10] initially developed the original Erd6s-Rényi la w to a stationary Gaussian sequence under a condition on the correlation function.M ore precisely, supp o se {e; ; j=1, 2, ••• } is a stationary G aussian sequence w ith Eefor some / 3>0 and 0<a 2 -=1+2 E r 1 .1=1 th e n fo r each 0<c<00 lim D(n , [c log n]).= a -V 2/ c, a. s.. O ur object o f this paper is to im prove Deo's result a n d o b ta in a g e n e ra l fo rm o f th e Erd6s-Rényi la w fo r stationary G aussian sequences.O ur re su lt is a s follows : Let fei ; j=1, 2,