A Probabilistic Proof of Stein's factors

We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalising puzzle in using Markov chain knowledge to view the celebrated Stein-Chen method for Poisson approximations. This paper complements the work of Barbour (1988) for the case of Poisson random variable approximation. The Stein-Chen method was introduced in Chen (1975) to estimate upper bounds for the total variation distance between the distribution of a sum of dependent 0 ? 1 random variables and a Poisson distribution. One way of understanding the Stein-Chen method is by introducing a Markov birth and death process whose equilibrium distribution is the approximating Poisson distribution, and using the properties of the Markov chains to assist in the estimation of the distance, as implemented in Barbour (1988) and Barbour and Brown (1992) see also Barbour, Holst and Janson (1992)]. The Markov birth and death process is an immigration death process, with immigration occurring according to the mean of the approximating Poisson distribution, and linear death rates. More precisely, the generator of the birth and death process is The corresponding Stein's equation of A for a bounded function f on IN is then given by A h(n) = f(n) ? Po()(f); (2) where Po() denotes a Poisson distribution with mean and (f) := R fdd: It is well-known that the solution of (2) is given by h(f; n) = ? Z 1 0 IE n f(Z(t)) ? Po()(f)]dt; where Z is a birth and death process with generator A , and IE n is the conditional expectation given Z(0) = n: We shall omit f in h(f; n) if there is no confusion.