Poisson Perturbations Poisson Perturbations Poisson Perturbations *

Stein's method is used to prove approximations in total variation to the distributions of integer valued random variables by (possibly signed) compound Poisson measures. For sums of independent random variables, the results obtained are very explicit, and improve upon earlier work of Kruopis (1983) and Cekanavicius (1997); coupling methods are used to derive concrete expressions for the error bounds. An example is given to illustrate the potential for application to sums of dependent random variables. ESAIM: Probability and Statistics October 1999, Vol. 3, p. 131–150 URL: http://www.emath.fr/ps/ POISSON PERTURBATIONS ∗ Andrew D. Barbour and Aihua Xia Abstract. Stein’s method is used to prove approximations in total variation to the distributions of integer valued random variables by (possibly signed) compound Poisson measures. For sums of independent random variables, the results obtained are very explicit, and improve upon earlier work of Kruopis (1983) and Čekanavičius (1997); coupling methods are used to derive concrete expressions for the error bounds. An example is given to illustrate the potential for application to sums of dependent random variables. Stein’s method is used to prove approximations in total variation to the distributions of integer valued random variables by (possibly signed) compound Poisson measures. For sums of independent random variables, the results obtained are very explicit, and improve upon earlier work of Kruopis (1983) and Čekanavičius (1997); coupling methods are used to derive concrete expressions for the error bounds. An example is given to illustrate the potential for application to sums of dependent random variables. Résumé. On utilise la méthode de Stein pour approximer, par rapport à la variation totale, la distribution d’une variable aléatoire aux valeurs entières par une mesure (éventuellement signée) de Poisson composée. Pour les sommes de variables aléatoires indépendantes, on obtient des résultats très explicites ; les estimations de la précision de l’approximation, construites à l’aide de la méthode de “coupling”, sont plus exactes que celles de Kruopis (1983) et de Čekanivičius (1997). Un exemple sert à illustrer le potentiel de la méthode envers les sommes de variables aléatoires dépendantes. AMS Subject Classification. 62E17, 60G50, 60F05. Received January 28, 1999. Revised July 26, 1999.