Le Cam's Inequality and Poisson Approximations

Abstract

where A = P1 + P2 + * .. + Pn Naturally, this inequality contains the classical Poisson limit law (Just set pi = A/n and note that the right side simplifies to 2A2/n), but it also achieves a great deal more. In particular, Le Cam's inequality identifies the sum of the squares of the pi as a quantity governing the quality of the Poisson approximation. Le Cam's inequality also seems to be one of those facts that repeatedly calls to be proved-and improved. Almost before the ink was dry on Le Cam's 1960 paper, an elementary proof was given by Hodges and Le Cam [18]. This proof was followed by numerous generalizations and refinements including contributions by Kerstan [19], Franken [15], Vervatt [30], Galambos [17], Freedman [16], Serfling [24], and Chen [11, 12]. In fact, for raw simplicity it is hard to find a better proof of Le Cam's inequality than that given in the survey of Serfling [25]. One purpose of this note is to provide a proof of Le Cam's inequality using some basic facts from matrix analysis. This proof is simple, but simplicity is not its raison d'etre. It also serves as a concrete introduction to the semi-group method for approximation of probability distributions. This method was used in Le Cam [20], and it has been used again most recently by Deheuvels and Pfeifer [13] to provide impressively precise results. The semi-group method is elegant and powerful, but it faces tough competition, especially from the coupling method and the Chen-Stein method. The literature of these methods is reviewed, and it is shown how they also lead to proofs of Le Cam's inequality.