Coupling and Poisson Approximation

We give an overview of the Stein{Chen method for establishing Poisson approximations of various random variables. Couplings of certain variables are used to gives explicit bounds for the total variation distance between the distribution of a random variable and a Poisson variable. Some applications are given. In some cases, explicit couplings may be used to obtain good estimates; in other applications it suuces to show the existence of couplings with certain monotonicity properties. 1. The Stein{Chen method Consider a random variable W that can be written as a sum P 2? I of 0{1 random variables, where ? is a nite index set. The variables I may be dependent, but we will be interested in cases where the dependencies are rather weak and the individual probabilities P(I = 1) rather small. It is then reasonable to try to approximate the distribution of W by a Poisson distribution. The Stein{Chen method is a powerful tool to justify such approximations by establishing an upper bound to the error, measured as the total variation distance between L(W) and the Poisson distribution Po() with a suitably chosen mean. In many cases, this method can be fruitfully combined with coupling methods to yield simple upper bounds. Furthermore, the bounds often turn out to be of the right order of magnitude. The method was originally formulated for normal approximation by Stein in 1970, but the basic idea applies also to other cases. The Poisson approximation version was worked out by Chen (1975). See also Stein (1986). The method has successfully been applied to a number of diierent problems by various authors. The coupling approach described here is developed in detail in Barbour, Holst and Janson (1992), to which we refer for proofs, further details, additional references, and many more applications. The central idea in the Stein{Chen method is that for any > 0 and A Z + there exists a function g ;A on Z + satisfying Stein's equation g ;A (j + 1) ? jg ;A (j) = I(j 2 A) ? Po()fAg; j 0: (1) Consequently, substituting W, taking expectations and interchanging the sides, we get The function g ;A can be given explicitly, but the method ignores the exact form of g ;A and uses only some simple estimates. In particular, we will here only use the estimate