On Hipp's compound Poisson approximations via concentration functions

Bero Roos

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This paper is devoted to a refinement of Hipp's method in the compound Poisson approximation to the distribution of the sum of independent but not necessarily identically distributed random variables. Approximations by related Kornya-Presman signed measures are also considered. By using alternative proofs, we show that several constants in the upper bounds for the Kolmogorov and the stop-loss distances can be reduced. Concentration functions play an important role in Hipp's method. Therefore, we provide an improvement of the constant in Le~Cam's bound for concentration functions of compound Poisson distributions. But we also follow Hipp's idea to estimate such concentration functions with the help of Kesten's concentration function bound for sums of independent random variables. In fact, under the assumption that the summands are identically distributed, we present a smaller constant in Kesten's bound, the proof of which is based on a slight sharpening of Le Cam's version of the Kolmogorov-Rogozin inequality.

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Bernoulli Volume 11, Number 3 (2005), 533-557.

First available in Project Euclid: 5 July 2005

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compound Poisson approximation concentration functions explicit constants Hipp's method individual risk model Kolmogorov distance Kornya-Presman signed measures random sums stop-loss distance sums of independent random variables upper bounds


Roos, Bero. On Hipp's compound Poisson approximations via concentration functions. Bernoulli 11 (2005), no. 3, 533--557. doi:10.3150/bj/1120591188.

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