Solution of Monge-Kantorovich Problem for a Certain Class
of Functionals and Application to Poisson Approximation

Rouzankin, Pavel (Novosibirsk, Russia)
ruzankin@math.nsc.ru

Let P and Q be Borel probability measures on the real line. Consider the functional $$\rho(t,P,Q)=\inf_{\xi,\eta}\{{\mathsf P}(r(\xi,\eta)\gt t)\},$$ where infimum is taken over all random variables $\xi$ and $\eta$ defined on a common probability space with distributions P and Q , respectively. R.M. Dudley proved (1968) that

$$\rho(t,P,Q)= \sup\{P(Z)-Q(Z_t): Z\mbox{ is closed in }U\},$$

where $Z_t=\{y\in U:r(Z,y)\le t\}$. The main goal of the talk is to give some more convenient representation for the functional $\rho$. Denote by F and G the distribution functions for P and Q , respectively. Then

$$\rho(t,P,Q)=\lim_{y\to\infty}S(y)-1=\lim_{y\to\infty}T(y)-1,$$

where functions S and T are defined by the requirement of left-continuity and by the relations:

$$\lim_{y\to-\infty}S(y)=\lim_{y\to-\infty}T(y)=0,$$

$$dS(y)=\max\{dF(y),T(y+dy-t)-S(y)\},$$

$$dT(y)=\max\{dG(y),S(y+dy-t)- T(y)\}$$

for all y . The functions S and T meeting the conditions above always exist and are uniquely defined.

The result is applied to estimate distance $\rho$ between binomial distribution and Poisson distribution.