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Limit Theorems for Stochastic Processes with Independent Increments

A. V. Skorokhod

Moscow

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 $\mathbf J_1$-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:
$$\xi_n(t)=\sum_{i\leq t(n+1)}\xi_{n,i},\quad\eta_n(t)=\sum_{i\leq t(n+1)}\eta_{n,i}.$$
Further, let distributions $\xi_n(t)$ and $\eta_n (t)$ converge to the distribution $\xi_0(t)$ for all $t$.
Then, the Levy 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
$$\lim_{\delta\to0}\sup_{\sup\limits_t|x(t)-y(t)|\leq\delta}|f(x(t))-f(y(t))|=0.$$


Received: 11.01.1957

 English version: Theory of Probability and its Applications, 1957, 2:2, 138–171