Weak convergence of sequences of first passage processes and applications

Suppose {Xn}n⩾1 are stochastic processes all of whose paths are nonnegative and lie in the space of right continuous functions with finite left limits. Moreover, assume that Xn (properly normalized) converges weakly to a process X, i.e., for some deterministic function μ and θn → 0, θn−1(Xn − μ) →dX. aper considers the description of the weak limiting behavior of the sequence of first passage processes X̃−1n(t) = inf{x : X̃n(x) ⩾ t} where X̃n(x) = ϱ(nx)Xn(x) and ϱ(·) is such that X̃n(x) has nondecreasing paths. We present a number of important motivating examples including empirical processes associated with U-statistics, empirical excursions above a given barrier, stopping rules in renewal theory and weak convergence in extreme value theory and point out the wide applicability of our result. Weak functional limit theorems for general quantile-type processes are derived. In addition, we investigate the asymptotic behavior of integrated kernel quantiles and establish: (i) an invariance principle; (ii) a strong law of large numbers; and (iii) a Bahadur-type representation which has many consequences, among which is a law of the iterated logarithm.