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Weak convergence of first passage time processes

Published online by Cambridge University Press:  14 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
Yale University
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Let D = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic process X in D, let the associated supremum process be S(X), where for any x ∊ D. It is easy to verify that S: D → D is continuous in any of Skorohod's (1956) topologies extended from D[0,1] to D[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergence Xn ⇒ X in D implies weak convergence S(Xn ) ⇒ S(X) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

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Type
Short Communications
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Journal of Applied Probability , Volume 8 , Issue 2 , June 1971, pp. 417 - 422
Copyright
Copyright © Applied Probability Trust 1971 

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