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Poisson approximation for a sum of dependent indicators: an alternative approach

Part of: Limit theorems

Published online by Cambridge University Press:  01 July 2016

N. Papadatos  and
V. Papathanasiou
N. Papadatos*
Affiliation:
University of Athens
V. Papathanasiou*
Affiliation:
University of Athens
*
∗ Postal address: Section of Statistics and Operational Research, Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece.
∗ Postal address: Section of Statistics and Operational Research, Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece.
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Abstract

The random variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Keywords

Totally negatively dependent indicatorChen-Stein equationnegatively related indicatorPoisson approximationtotal variation distancebirthday problemmonotone coupling

MSC classification

Primary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 609 - 625
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research partially supported by the research foundation of the University of Athens.

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