Weak convergence of first passage time processes

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 X n ⇒ X in D implies weak convergence S ( X n ) ⇒ S ( X ) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

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