On the rate of Poisson convergence

Abstract

Let X1, …, Xn be independent Bernoulli random variables, and let pi = P[Xi = 1], λ = Σi=1n pi and Σi=1n Xi. Successively improved estimates of the total variation distance between the distribution ℒ(W) of W and a Poisson distribution Pλ with mean λ have been obtained by Prohorov[5], Le Cam [4], Kerstan[3], Vervaat[8], Chen [2], Serfling[7] and Romanowska[6]. Prohorov, Vervaat and Romanowska discussed only the case of identically distributed Xi's, whereas Chen and Serfling were primarily interested in more general, dependent sequences. Under the present hypotheses, the following inequalities, here expressed in terms of the total variation distancewere established respectively by Le Cam, Kerstan and Chen:(Kerstan's published estimate of ([3], p.174, equation (1)) is a misprint for , the constant 2·1 appearing twice on p. 175 of his paper.) Here, we use Chen's [2] elegant adaptation of Stein's method to improve hte estimates given in (1·1), and we complement these estimates with a reverse inequality expressed in similar terms. Second order estimates, and the case of more general non-negative integer valued X's, are also discussed.

On a relationship between Uspensky's theorem and poisson approximations

“…A more elaborate evaluation of the total variation distance, considering terms up to a4, gives which is asymptotically sharper than the bound in Barbour and Hall (1984). Comparison with Barbour's results (1987) shows that in the range where 2 is large the above formulae are more useful, while when 2 remains bounded we need higher order terms in order to get the same order of accuracy (e.g., if p i ( n ) -2/n, then (3.19) requires terms of order ?/-m+l to achieve an error of o r d e r n-(m+l)/~).…”

“…Pfeifer (1983Pfeifer ( , 1985, Barbour and Hall (1984), Deheuvels and Pfeifer (1986a, 1986b, 1987, Barbour (1987)), using e.g. Stein's method (Barbour and Hall (1984), Barbour (1987)) or a Poisson convolution semigroup approach on a suitable Banach space, generalizing an operator-theoretic approach originally due to LeCam (1960).…”

Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.

“…The result is an application of a theorem of Barbour and Hall [2]. Their theorem states that if A := j I j is the sum of independent indicator random variables indexed by j and…”

The median of the poisson distribution

“…Many authors have obtained sharp estimates for such a distance by using different approaches, such as SteinChen method (cf. Chen [9], Barbour and Hall [5] and Barbour et al [6]), semigroup techniques (cf. Deheuvels and Pfeifer [11] and Deheuvels et al [12]), and Charlier expansions (cf.…”

“…. ., 5) and U and V are independent identically distributed random variables having the uniform distribution on [0, 1]. Here and hereafter, all of the random variables appearing under the same expectation sign are supposed to be mutually independent.…”