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

Let €(t),…» €n(t)»… and (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 ƒ(€(t)) converges to the distribution of ƒ(€ (t)) as ƒ i->oo for any functional ƒ 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 € (t) T (0 < t < 1) without discontinuities of the second kind. For a set T» {t₁, C [0, 1] the metric pT is defined on as in 2.3. The metric pT defines on the X the minimal topology in which all functionals continuous in Skorohod’s metric and also the functionals x(t₁ - 0), x(t₁),…»x(t_n - 0), *(t_n),… are continuous. We will give necessary and sufficient conditions under which the distribution of ƒ(€_n(t)) converges to the distribution of ƒ (€ (t)) as n-> oo for any completely continuous functional ƒ, i.e. for any functional ƒ which is continuous in any of the metrics p_T defined in 2.3.