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Kerstan’s method for compound Poisson approximation. (English) Zbl 1041.62011
Summary: We consider the approximation of the distribution of the sum of independent but not necessarily identically distributed random variables by a compound Poisson distribution and also by a finite signed measure of higher accuracy. Using J. Kerstan’s method [Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 173–179 (1964; Zbl 0123.35403)], some new bounds for the total variation distance are presented.
Recently, several authors had difficulties applying Stein’s method to the problem given. For instance, A. D. Barbour, L. Y. H. Chen and W.-L. Loh [Ann. Probab. 20, 1843–1866 (1992; Zbl 0765.60015)] used this method in the case of random variables on the nonnegative integers. Under additional assumptions, they obtained some bounds for the total variation distance containing an undesirable log term. In the present paper, we shall show that Kerstan’s approach works without such restrictions and yields bounds without log terms.

MSC:
62E17 Approximations to statistical distributions (nonasymptotic)
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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