Compound Poisson approximation with association or negative association via Stein's method

Fraser Daly

doi:10.1214/ecp.v18-2751

Abstract

In this note we show how the properties of association and negative association can be combined with Stein's method for compound Poisson approximation. Applications include $k$--runs in iid Bernoulli trials, an urn model with urns of limited capacity and extremes of random variables.

Highlights

Introduction and main resultsIn recent years Stein’s method has proved to be an effective technique for probability approximation, often yielding explicit error bounds and working well in the presence of dependence
In this note we show how the properties of association and negative association can be combined with Stein’s method for compound Poisson approximation
Our purpose in this note is to show how assumptions of association or negative association may be combined with Stein’s method in a compound Poisson approximation setting. This provides an analogue of the idea of a ‘monotone coupling’ in Stein’s method for Poisson approximation

Summary

Introduction and main results

In recent years Stein’s method has proved to be an effective technique for probability approximation, often yielding explicit error bounds and working well in the presence of dependence. Stein’s method may be applied in a wide variety of settings: in this note we consider compound Poisson approximation. Our purpose in this note is to show how assumptions of association or negative association may be combined with Stein’s method in a compound Poisson approximation setting. The same is true if we make assumptions of association or negative association in a compound Poisson approximation setting, as will be demonstrated in the applications of Section 2. Compound Poisson approximation with association or negative association for all non–decreasing functions f and g. Xj , j∈J (i) and Wi = W − Xi − Zi. As in the work of Barbour et al [1] or Roos [13], we define our approximating compound Poisson random variable U by setting.

Stein’s method for compound Poisson approximation

Main results

Independent summands

An urn model with overflow

Extremes