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Nonnormal Small Jump Approximation of Infinitely Divisible Distributions

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  22 February 2016

Zhiyi Chi
Zhiyi Chi*
Affiliation:
University of Connecticut
*
∗ Postal address: Department of Statistics, 215 Glenbrook Road, U-4120, Storrs, CT 06269, USA. Email address: zhiyi.chi@uconn.edu
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Abstract

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We study a type of nonnormal small jump approximation of infinitely divisible distributions. By incorporating compound Poisson, gamma, and normal distributions, the approximation has a higher order of cumulant matching than its normal counterpart, and, hence, in many cases a higher rate of approximation error decay as the cutoff for the jump size tends to 0. The parameters of the approximation are easy to fix, and its random sampling has the same order of computational complexity as the normal approximation. An error bound of the approximation in terms of the total variation distance is derived. Simulations empirically show that the nonnormal approximation can have a significantly smaller error than its normal counterpart.

Keywords

Infinitely divisiblenormal approximationcompound Poisson approximationgamma approximationcumulant matchingsampling

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G51: Processes with independent increments; L'evy processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 963 - 984
Copyright
© Applied Probability Trust 

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