Multivariate normal approximations by Stein's method and size bias couplings

Abstract

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve e.g., martingale, Markov chain, or various mixing assumptions.

Total Variation Distance for Poisson Subset Numbers

“…Nevertheless, the upper bound, Bound (9), can be easily calculated in all cases. The total variation distance (8), and hence any upper bound to it, may naturally exceed the difference between the true and approximated probabilities at any particular value. We note, however, that the total variation and its bound are nevertheless of a similar order of magnitude as the observed differences, and, moreover, that the bound on TV is not far from the actual value.…”

“…First, choose an index α with probability proportional to p α . Then replace X α by a variable X α α having the X α size biased distribution, and which is independent of X β , β = α (see, for example, [8].) For a sum of dependent variables, the procedure is nearly the same.…”

“…Hence, to size bias a random variable X ∈ {0, 1} with P (X = 1) > 0, we must have X * = 1. For discussion of the relation between size biasing and Stein's method, and further constructions such as the one we will apply below, see Goldstein and Rinott [8] and Goldstein and Reinert [7]. We can restate the characterizing equation (5) for the Poisson in terms of size biasing, using equation (6), as follows: Z has a Poisson distribution if and only if Z * = Z + 1, in distribution, or equivalently, if and only if Z * − 1 and Z have the same distribution.…”

Spin glasses and Stein’s method

“…A number of techniques for carrying out this step are available in the literature, e.g. exchangeable pairs [27], diffusion generators [4], dependency graphs [2,3], size bias couplings [16], zero bias couplings [15], couplings for Poisson approximation [5,11], specialized procedures like [19,22,23], and some recent advances [7][8][9][10]. Incidentally, Stein's method was applied to solve a problem in the interface of statistics and spin glasses in [6].…”

Concentration of measures via size-biased couplings

“…For a given nonnegative random variable Y with finite nonzero mean µ, recall (see [14], for example) that Y s has the Y -size biased distribution if…”