Probability and Mathematical Statistics: Vol. 3, Fasc. 2

ON THE CENTRAL LIMIT THEOREM IN BANACH SPACE $c0$

Vygantas Paulauskas

Abstract: In the paper the central limit theorem and the rates of convergence in this theorem in Banach space $c0$ are considered. Let $(1) (n) qi = qi ,...,qi ,...), i = l,2,...,$ be i.i.d. $c0$ -valued random variables with $Eq1 = 0$ and covariance matrix $T$ . Let $m$ be a zero-mean Gaussian measure on $c0$ with covariance matrix $T$ ,

n Fn(A) = P(n- 1/2 sum qi (- A). i=1

The main result of the paper can be formulated as follows: if $(j) -1/2 |q1 |< Mj = (ln j) aj,j > j0,$ where $(aj)$ is an arbitrary sequence of positive numbers tending to zero, then $Fn$ converges weakly to $m$ . Moreover, if instead of $aj$ we take a slowly increasing sequence $1/2+e (lnkj) ,$ where $lnkx = ln lnk-1x$ and $k > 2$ is an arbitrary integer, then it is possible to construct $qi, i > 1,$ failing the central limit theorem.

If $(j) 2 (j)2 -(1+d) |q1 |< M sj, sj = E(q1 ) = (lnj) , j > 2, d > 0,$ and $T$ satisfies one additional condition, then we get the estimate

-1/2+e sur>p0 |Fn(||x||< r)- m(||x||< r) = O(n ), e > 0.

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

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