Strong Limit Theorems for Stochastic Processes and Orthogonality Conditions for Probability Measures

Abstract

Let x (t), 0≦t≦T, be the Wiener process, that is, a real Gaussian stochastic process with

$${{E}_{x}}\left( t \right)=0,Ex\left( t \right)x\left( S \right)=R\left( t,s \right)=\min \left\{ t \right.,\left. s \right\}$$

(1)

and let N _n for n = 1, 2,... be the sequence of increasing integers. Consider the following functional of x (t):

$${{U}_{n}}\left[ x\left( t \right) \right]={{\sum\limits_{k=1}^{{{N}_{n}}}{\left[ x\left( \frac{kT}{{{N}_{n}}} \right)-x\left( \frac{\left( k-1 \right)T}{{{N}_{n}}} \right) \right]}}^{2}}.$$

(2)

In 1940 Lévy [1] discovered the following interesting result concerning this functional.