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Stopping times and tightness. II. (English) Zbl 0686.60036
In his Cambridge thesis of 1977 the author established sufficient conditions for sequences of stochastic processes to converge in the sense of weak convergence. Part of this work has been published in part I of the present paper, ibid. 6, 335-340 (1978; Zbl 0391.60007). Another part appeared in print rather late. In this paper it is shown that for a sequence $$(X_ n)$$ of random functions in the Skorokhod space D to converge weakly to $$X_{\infty}$$ the following conditions are sufficient:
(a) $$(X_ n)$$ converges to $$X_{\infty}$$ via finite-dimensional distributions, (b) $$X_{\infty}$$ is continuous, and (c) for each $$\epsilon$$ and L there exists a $$\delta >0$$ such that $\lim_{n\to \infty}\inf \Gamma_{X_ n}(L,\epsilon,\delta)>0,$ where for a random function X in D, $$\Gamma_ X$$ is defined as the supremum of $$\Gamma\geq 0$$ satisfying the inequalities $P(X(T+\delta ')-X(T)\leq \epsilon | T\leq L)\geq \Gamma,\quad and$
$P(X(T+\delta ')-X(T)\geq -\epsilon | T\leq L)\geq \Gamma \quad for\quad all\quad 0<\delta '<\delta,$ whenever T is a stopping time for X with P(T$$\leq L)\geq \epsilon$$. Applications to semimartingale sequences are included.
Reviewer: H.Heyer

##### MSC:
 60G44 Martingales with continuous parameter 60B10 Convergence of probability measures
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