DONSKER'S THEOREM FOR DISCRETIZED DATA

Abstract

Inspired by Doob’s suggestion in 1949, Donsker (1952) proved the Kolmogorov-Smirnov theorem in an elegant way via the functional central limit theorem (the invariance principle). In that theorem, the underlying distribution is assumed to be a continuous distribution. On the other hand, real data in practice is always given in a discretized (rounded) form. In this paper, we establish an invariance principle for discretized data in the fashion of the modern empirical process theory to obtain a (right) Kolmogorov-Smirnov test for discretized data. To illustrate our problem let us begin with the most basic example. We denote by F0 the uniform distribution on [0, 1]. Let {X1, . . . , Xn} be an independent sequence of [0, 1]-valued random variables with the common law F0. Set δn = 0.01. Suppose that we can actually observe the data {X i } which is discretized (rounded) up to δn: X1 = 0.67774205 X 1 = 0.68 X2 = 0.81124449 X 2 = 0.81 · · · Xn = 0.61694806 X n = 0.62. We denote by F̂n and F̂ n the empirical distribution functions of {X1, . . . , Xn} and {X 1 , . . . , X n}, respectively. Then, the Kolmogorov-Smirnov statistic Dn = sup t∈[0,1] n|F̂n(t) − F0(t)| converges in distribution to supu∈[0,1] |B◦(u)|, where u ❀ B◦(u) is a standard Brownian bridge. On the other hand, as we will show below, if δn = o(n −1/2), the test statistic D n = sup t∈[0,1] n|F̂ n(t) − F0(t)|