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1996). On an estimate for the concentration function of sums of independent random variables. Theor. Probab. Appl. 41, 560-569. PETROV, V. V. (1976). Limit theorems for sums of independent random variables. Springer, Berlin. PETROV, V. V. (1995). Limit theorems of probability theory. Oxford University Press, New York. POSTNIKOVA, L. , AND YUDIN, A. A. (1978). A sharper form of the inequality for the concentration function. Theor. Probab. Appl. 23, 359362. ROGOZIN, B. A. (1961a). An estimate for concentration functions.

We will deal later with this term by means of the Paley-Zygmund argument. Proof. 21) follows from Holder. 21) for m = 1 and we can proceed by induction. 1. • m-l} im 1 (i) C {1, ... 's him in loo(Lr). 24) 26 E. Gine, R. Latala, and J. 24) complete the induction step. > 1). 6, we apply Paley-Zygmund as above, but now with r < p replacing 1 < p. 7. There is a constant Kr,p,m such that for 0 m < 00, and hi ~ 0 with integrable p-th powers, [EJ ~~( EJc J~{l, ... ,p,m{ 2*~ + L [EJ"l~(EJ«p;rt·l}. J~{l, ...

14) Q(F, b) ~ cb ! Itl::5b- 1 IF(t) Idt. 15) ~ cb J Ip(t) In dt. 16) Ipet) In(l-a) dt, if no: is integer and b 2:: E IXI 3 /(72. 7). Our proofs are based on non-uniform estimates in the Central Limit Theorem (CLT) and on elementary properties of concentration functions (see the proof of Lemma 1 in GZ (1998) and Zaitsev (1987, 1992)). In this respect our proofs differ from most of the previous papers, where Esseen's (1968) method of characteristic functions had been extensively used. One should note however that the CLT approach was applied in the seminal paper of Kolmogorov (1958).

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