A Note on the Dobrushin - Dudley Theorem and Poisson Approximation of Empirical Processes

Borisov, Igor (Novosibirsk, Russia)
SIBAM@math.nsc.ru

Let $({\cal S},\, {\cal A})$ be an arbitrary measurable space. Denote by $\Sigma=\Sigma ({\cal A}\times {\cal A},\,D)$ the minimal $\sigma$-field generated by the standard product $\sigma$-field ${\cal A}\times {\cal A}$and diagonal $D=\{(x,y): \, x=y\}$ of the product space.

Denote by $V({\cal L}(X), {\cal L}(Y))$ the total variation distance between distributions of random variables X and Y taking values in $({\cal S},\, {\cal A})$.We discuss the following generalization of the fundamental result proved independently by R.M.Dudley (for Polish spaces, 1968) and by R.L. Dobrushin (for separable metric spaces, 1970).

Theorem 1. On the probability space $({\cal S}\times{\cal S},\,\Sigma)$,one can construct copies X₀ and Y₀ of arbitrary random variables X and Y in $({\cal S},\, {\cal A})$ such that  $$V({\cal L}(X), {\cal L}(Y))={\mathsf P}(X_0\neq Y_0),$$

and, moreover,

 $$V({\cal L}(X), {\cal L}(Y))=\inf_{X,Y}{\mathsf P}(X\neq Y),$$

where the infimum in (2) is taken over all pairs (X, Y) based on the above probability space and having initial marginal distributions.

Let $\{X_{nk};\,k\le n\},\,n=1,2,...,$ be row-wise i.i.d.r.v-s in $({\cal S}, {\cal A}).$ For each n, introduce the normalized empirical process based on the sample $\{X_{nk};\,k\le n\}$ and indexed by a family ${\cal F}$ of real-valued Borel measurable functions on $({\cal S}, {\cal A}):$

$$S_n(f):=\sum_{k\le n}f(X_{nk}),\,\,\,f\in \cal F.$$

We consider S_n as a r.v. in the measurable space $(R^{\cal F}, {\cal C})$of all real-valued functions indexed by ${\cal F}$, with the cylinder $\sigma$-field ${\cal C}$ (Kolmogorov's space). As a consequence of the theorem above and the classical Prohorov - Le Cam estimates for the total variation distance in the Poisson theorem one can obtain the following result.

Theorem 2. Let ${\lambda}$ be a finite measure on ${\cal A}$ such that $V(n{\cal L}(X_{n1}),\lambda)\to 0$ as $n\to \infty.$ If, additionally,

$$\lim_{n\to \infty}{\bf Pr}^*\biggl\{ \bigcup_{f\in {\cal F}} \{f(X_{n1})\ne 0\} \biggr\}= 0,$$

where ${\bf Pr}^*$ is the outer probability, then one can construct on a common probability space random process S_n and the generalized Poisson process

$$\Pi (f):=\int\limits_{\cal S}f(x)\pi_{\lambda}(dx),$$

where $\pi_{\lambda}$ is a Poisson point process on ${\cal A}$ with mean measure $\lambda,$ such that $$\lim_{n\to\infty}{\bf Pr}\biggl\{ \bigcup_{f\in {\cal F}}\{S_n(f)\ne \Pi (f)\} \biggr\} =0.$$