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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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