Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables

Abstract:

Let ${\xi _1}(t), \ldots ,{\xi _n}(t), \ldots$ and $\xi (t)$ be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$ as $n \to \infty$ for any functional f continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions $x(t)(0 \leqslant t \leqslant 1)$ without discontinuities of the second kind. For a set $T = \{ {t_1}, \ldots {t_n}, \ldots \} \subset [0,1]$ the metric ${\rho _T}$ is defined on X as in 2.3. The metric ${\rho _T}$ defines on the X the minimal topology in which all functional continuous in Skorohod’s metric and also the functional $x({t_1} - 0),x({t_1}), \ldots ,x({t_n} - 0),x({t_n}), \ldots$ are continuous. We will give necessary and sufficient conditions under which the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$ as $n \to \infty$ for any completely continuous functional f, i.e. for any functional f which is continuous in any of the metrics ${\rho _T}$ defined in 2.3.