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December, 1975 On Vague Convergence of Stochastic Processes
R. V. Erickson, Vaclav Fabian
Ann. Probab. 3(6): 1014-1022 (December, 1975). DOI: 10.1214/aop/1176996227


Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. The proof we use is simple and of interest even when other proofs are available.


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R. V. Erickson. Vaclav Fabian. "On Vague Convergence of Stochastic Processes." Ann. Probab. 3 (6) 1014 - 1022, December, 1975.


Published: December, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0365.60037
MathSciNet: MR385951
Digital Object Identifier: 10.1214/aop/1176996227

Primary: 60B10
Secondary: 60G99 , 62E20

Keywords: Skorohod metric , stochastic process , tightness , uniform metric , vague convergence

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 6 • December, 1975
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