# Bonferroni Inequalities and Deviations of Discrete Distributions

#### Abstract

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1]. BONFERRONI INEQUALITIES; PROBABILITY GENERATING FUNCTIONS AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60E15 SECONDARY 60E10 Let A,, i e H be a finite family of arbitrary events in an arbitrary probability space (0, 4, P). Define N= Ci eH I(A), where I( ) denotes the indicator of the event in parentheses. This is the (random) number of events that occur. For r ? 1 let S,= EMCH, IMI=r, P('ieM Ai) (the void product is defined as 0, thus So = 1). It is well known that S,.= E(N), the rth binomial moment of N. In general, estimates of the form (1) P(N=k)<(>) I c,.S, or P(N!k)?(?) I cS, rO0 r>O are called Bonferroni-type inequalites. Such inequalities are applied widely in practice, when the events in questions are dependent, with a complicated dependence structure, but the importance of the event {N= 0} requires good estimations for its probability. A convincing example is where the individual events represent the exceedances of a certain level by different random variables (the sizes of shocks, for instance). In many cases we also need to know, or to approximate, the whole distribution of N: then Bonferroni inequalities can again help. For instance, by using Bonferroni inequalities it is easy to prove that N converges in distribution to the Poisson law with parameter X > 0 if I HI --+00 and the events in consideration vary in such a way that S,--+Zlr! for each fixed r ?0. This is the case when the events are all independent and become asymptotically negligible. A large variety of Bonferroni-type inequalities together with methods for constructing Received 1 August 1994. * Postal address: Department of Probability Theory and Statistics, E6tv6s Lorind University, Budapest, Mi'zeum krt.6-8, H-1088 Hungary. E-mail: moritamas@ludens.elte.hu Research supported by the Hungarian National Foundation for Scientific Research, Grant N' 1405.

### Cite this paper

@inproceedings{Mri2008BonferroniIA, title={Bonferroni Inequalities and Deviations of Discrete Distributions}, author={Tam{\'a}s F. M{\'o}ri and TAMAS E. MORI}, year={2008} }