For all @math , we show that the set of Poisson Binomial distributions on @math variables admits a proper @math -cover in total variation distance of size @math , which can also be computed in polynomial time. We discuss the implications of our construction for approximation algorithms and the computation of approximate Nash equilibria in anonymous games.
In Probability and Statistics there is a broad literature studying various properties of these distributions; see @cite_18 for an introduction to some of this work. Many results provide approximations to the Poisson Binomial distribution via simpler distributions. In a well-known result, Le Cam @cite_19 shows that, for any vector @math , @math where @math is the Poisson distribution with parameter @math . Subsequently many other proofs of this bound and improved ones, such as Theorem of , were given, using a range of different techniques; @cite_21 @cite_2 @cite_24 @cite_8 is a sampling of work along these lines, and Steele @cite_5 gives an extensive list of relevant references. Much work has also been done on approximating PBDs by Normal distributions (see e.g. @cite_25 @cite_11 @cite_13 @cite_27 @cite_23 ) and by Binomial distributions; see e.g. Ehm's result @cite_7 , given as Theorem of , as well as Soon's result @cite_4 and Roos's result @cite_22 , given as Theorem of .
@cite_18 @cite_4 @cite_22 @cite_7 @cite_8 @cite_21 @cite_24 @cite_19 @cite_27 @cite_23 @cite_2 @cite_5 @cite_13 @cite_25 @cite_11
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A binomial approximation theorem for dependent indicators using Stein's method and coupling is proved. The approximating binomial distribution B(n � ,p � ) is chosen in such a way that its first moment is equal to that of W and its variance is asymptotically equal to that of W as ntends to infinity where W is the sum of independent indicators and pis bounded away from 1. Three examples, one of which concerns two different approximations for the hypergeometric distribution, are given to illustrate applications of the theorem obtained. The Poisson binomial distribution is approximated by a binomial distribution and also by finite signed measures resulting from the corresponding Krawtchouk expansion. Bounds and asymptotic relations for the total variation distance and the point metric are given. Upper and lower bounds are given for the total variation distance between the distribution of a sum S of n independent, non-identically distributed 0-1 random variables and the binomial distribution (n, p) having the same expectation as S. The proof uses the Stein--Chen technique. Equivalence of the total variation and the Kolmogorov distance is established, and an application to sampling with and without replacement is presented. On utilise l'adaptation elegante de Chen de la methode de Stein pour ameliorer des estimateurs This paper introduces an estimate of approximation errors for the distribution function of a sum of random indicators. The approximation is demonstrated with the help of the problem of estimating the distribution function of the empty cells number in the equiprobable scheme for group distribution of particles. Preface.- 1.Introduction.- 2.Fundamentals of Stein's Method.- 3.Berry-Esseen Bounds for Independent Random Variables.- 4.L^1 Bounds.- 5.L^1 by Bounded Couplings.- 6 L^1: Applications.- 7.Non-uniform Bounds for Independent Random Variables.- 8.Uniform and Non-uniform Bounds under Local Dependence.- 9.Uniform and Non-Uniform Bounds for Non-linear Statistics.- 10.Moderate Deviations.- 11.Multivariate Normal Approximation.- 12.Discretized normal approximation.- 13.Non-normal Approximation.- 14.Extensions.- References.- Author Index .- Subject Index.- Notation. 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. The sum of finitely many variates possesses, under familiar conditions, an almost Gaussian probability distribution. This already much discussed "central limit theorem"(x) in the theory of probability is the object of further investigation in the present paper. The cases of Liapounoff(2), Lindeberg(3), and Feller(4) will be reviewed. Numerical estimates for the degrees of approximation attained in these cases will be presented in the three theorems of §4. Theorem 3, the arithmetical refinement of the general theorem of Feller, constitutes our principal result. As the foregoing implies, we require throughout the paper that the given variates be totally independent. And we consider only one-dimensional variates. The first three sections of the paper are devoted to the preparatory Theorem 1 in which the variates.meet the further condition of possessing finite third order absolute moments. Let X , Xi, • • • , Xn be the given variates. For each k k = ,2, ■ ■ ■ , n) let ^(Xk) and ixs Xk) denote, respectively, the second and third order absolute moments of Xk about its mean (expected) value a*. These moments are either both zero or both positive. The former case arises only when Xk is essentially constant, i.e., differs from its mean value at most in cases of total probability zero. To avoid trivialities we suppose that PziXk) >0 for at least one k (k = 1, 2, • • • , n). The non-negative square root of m Xk) is the standard deviation of Xk and will be denoted by ak. We call