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Weak convergence of sequences of first passage processes and applications

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  • Ralescu, Stefan S.
  • Puri, Madan L.

Abstract

Suppose {Xn}n[greater-or-equal, slanted]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 [mu] and [theta]n --> 0, . This paper considers the description of the weak limiting behavior of the sequence of first passage processes where and [varrho](·) is such that 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.

Suggested Citation

  • Ralescu, Stefan S. & Puri, Madan L., 1996. "Weak convergence of sequences of first passage processes and applications," Stochastic Processes and their Applications, Elsevier, vol. 62(2), pages 327-345, July.
    • Handle: RePEc:eee:spapps:v:62:y:1996:i:2:p:327-345
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    References listed on IDEAS

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    1. Veraverbeke, Noël, 1987. "A kernel-type estimator for generalized quantiles," Statistics & Probability Letters, Elsevier, vol. 5(3), pages 175-180, April.
    2. Shorack, Galen R., 1979. "Weak convergence of empirical and quantile processes in sup-norm metrics via kmt-constructions," Stochastic Processes and their Applications, Elsevier, vol. 9(1), pages 95-98, August.
    3. M. Falk, 1983. "Relative efficiency and deficiency of kernel type estimators of smooth distribution functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 37(2), pages 73-83, June.
    4. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
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