Limit Theorems for Stochastic Processes with Independent Increments

Abstract

The general results in [8] are used for the case of convergence of processes with independent increments. In particular the following results are obtained: 2.6. Theorem. Let the distributions of processes with independent increments $\xi _n (t)$ converge to the distribution of a continuous probability process with independent increments $\xi _0 (t)$ for all t. Then, there exists an $\bar x_n (t)$, such that the distribution $f(\xi _n (t) - \bar x_n (t))$ converges to the distribution $f(\xi _0 (t))$ if the functional f is continuous in the ${\bf J}_1 $-topology (see [8]). 3.4. Theorem. Let $\xi _{n,1} , \cdots ,\xi _{n,n} $ be independent random variables with, the same distributions, and also let $\eta _{n,1} , \cdots ,\eta _{n,n} $ be independent random variables with the same distributions: \[ \xi _n ( t ) = \sum\limits_{i \leqq t(n + 1)} {\xi _{n,i} } ,\qquad \eta _n (t) = \sum\limits_{i \leqq t(n + 1)} {\eta _{n,i} } . \] Further, let distributions $\xi _n (t)$ and $\eta _n (t)$ converge to the distribution $\xi _0 (t)$ for all t. Then, the Lévy distance between distribution functions of random variables $f(\xi _n (t))$ and $f(\eta _n (t))$ tends to zero as $n \to \infty $, for all functional f, such that \[ \mathop {\lim }\limits_{\delta \to 0} \mathop {\sup }\limits_{\mathop {\sup }\limits_t | {x(t) - y(t)} | \leqq \delta } | {f(x(t)) - f(y(t))} | = 0. \]