Best finite constrained approximations of one-dimensional probabilities

Abstract

This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.Keywords. Constrained approximation, best uniform approximation, asymptotically best approximation, Kantorovich distance, balanced function, quantile function.When endowed with the metric d r , the space P r is separable and complete, and d r (µ n , µ) → 0 implies that µ n → µ weakly. Note that P r ⊃ P s and d r ≤ d s whenever r < s. On P s , the metrics d r and d s are not equivalent, as the example of µ n = (1 − n −s )δ 0 + n −s δ n shows, for which d s (µ n , δ 0 ) ≡ 1, and yet lim n→∞ d r (µ n , δ 0 ) = 0 for all r < s, and hence µ n → δ 0 weakly. The reader is referred to [12,32] for details on the mathematical background of the Kantorovich distance, and to [17,32] for a discussion of its usefulness in the study of mass transportation and quantization problems.

Asymptotics of One-Dimensional Lévy Approximations

“…The first main result in this section asserts that nd ǫ (µ, δ un • ) does converge, to an easily determined limit, if µ −1 is absolutely continuous. The result is reminiscent of a theorem regarding best uniform d W -approximations [37,Thm.5.15] (see also [7,16]), but unlike in that theorem, no integrability assumption on dµ −1 /dλ is needed, and the limit in question always is finite. When formulating the result, it is helpful to use the function Ω :…”

“…, referred to as the inverse measure of µ; see, e.g., [6,37]. For convenience, write G Fµ and G F −1 µ simply as G µ and G µ −1 , respectively.…”

“…In a spirit similar to [18,37], the present article addresses the approximation problem relative to the classical Lévy metric,…”

“…An interesting variant of (1.4)-(1.6) considers best uniform approximations of µ ∈ P, that is, best approximations of µ by ν ∈ P * n , subject to the additional requirement that nν({x}) is a (positive) integer for every x ∈ supp ν. Best uniform (or, more generally, best constrained) approximations have recently attracted considerable interest, not least in view of potential applications in stochastic processes and differential equations [7,8,16,17,36,37]; they may also be viewed as deterministic analogues of (random) empirical measures [6,9,14]. With δ un • denoting a best uniform d-approximation of µ, trivially d(µ, δ •,n • ) ≤ d(µ, δ un • ).…”

“…With δ un • denoting a best uniform d-approximation of µ, trivially d(µ, δ •,n • ) ≤ d(µ, δ un • ). For µ being the standard normal distribution, [37,Ex.5.18] reports that…”