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

Abstract

Let ξ 1 ( t ) , … , ξ n ( t ) , … {\xi _1}(t), \ldots ,{\xi _n}(t), \ldots and ξ ( t ) \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 ( ξ n ( t ) ) f({\xi _n}(t)) converges to the distribution of f ( ξ ( t ) ) f(\xi (t)) as n → ∞ 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 ⩽ t ⩽ 1 ) x(t)(0 \leqslant t \leqslant 1) without discontinuities of the second kind. For a set T = { t 1 , … t n , … } ⊂ [ 0 , 1 ] T = \{ {t_1}, \ldots {t_n}, \ldots \} \subset [0,1] the metric ρ T {\rho _T} is defined on X as in 2.3. The metric ρ T {\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 ) , … , x ( t n − 0 ) , x ( t n ) , … 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 ( ξ n ( t ) ) f({\xi _n}(t)) converges to the distribution of f ( ξ ( t ) ) f(\xi (t)) as n → ∞ n \to \infty for any completely continuous functional f, i.e. for any functional f which is continuous in any of the metrics ρ T {\rho _T} defined in 2.3.