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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 7, Fasc. 1,
pages 27 - 58
 

LAW OF THE ITERATED LOGARITHM FOR SUBSEQUENCES

Allan Gut

Abstract: Let (S ) oo 
  n n=1  denote the partial sums of i.i.d. random variables with mean 0. The present paper investigates the quantity

lim supS  / V~ n--loglogn-,
 k-->o o   nk   k       k
where (n ) oo 
  k k=1  is a strictly increasing subsequence of the positive integers. The first results are that if EX2  <  oo ,
   1 then the limit superior equals s V~ 2 a.s. for subsequences which increase ”at most geometrically”, and se*, where
e* = inf(e > 0; sum  (log n )- e2/2 <  oo ),
             k      k
for subsequences which increase ”at least geometrically”. We also perform a refined analysis for the latter case and finally present criteria for the finiteness of

           V~ ----------2
E sukp(Snk/ nk log lognk)
in both cases.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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