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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 11, Fasc. 2,
pages 237 - 246
 

ON THE CENTRAL LIMIT THEOREM WITH ALMOST SURE CONVERGENCE

Peter Schatte

Abstract: Let S
 n  be the partial sums of i.i.d.r.v.’s with zero means and variance one and let a(x) be a real function. In this paper, sufficient conditions are given under which a(S / V~ n)
   n converges almost surely to  integral  oo  a(x)dP(x).
 - oo  Two variants of convergence are considered: limitation of       V~ --
a(Sn/  n) by logarithmic means and limitation of        V~ ---
a(Snk/  nk) by arithmetic means, where       ka
nk = c  , a > 0, c > 1. Under the same assumptions in the same sense,                 V~ --
a(max1<m <n Sm/  n) converges almost surely to    integral  oo 
2 0  a(x)dP(x).

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

Key words and phrases: -

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