Weak convergence of first passage time processes

Abstract

LetD = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic processXinD,let the associatedsupremum processbeS(X), wherefor anyx ∊ D. It is easy to verify thatS:D→Dis continuous in any of Skorohod's (1956) topologies extended fromD[0,1] toD[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergenceXn⇒XinDimplies weak convergenceS(Xn) ⇒S(X) inDby virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).