IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v62y1996i2p327-345.html
   My bibliography  Save this article

Weak convergence of sequences of first passage processes and applications

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(96)00054-3
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stilian A. Stoev & Murad S. Taqqu, 2007. "Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 55-87, March.
    2. Roussas, George G., 1995. "Asymptotic normality of a smooth estimate of a random field distribution function under association," Statistics & Probability Letters, Elsevier, vol. 24(1), pages 77-90, July.
    3. Furrer, Hansjorg & Michna, Zbigniew & Weron, Aleksander, 1997. "Stable Lévy motion approximation in collective risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 20(2), pages 97-114, September.
    4. Anatolii A. Puhalskii, 2003. "On Large Deviation Convergence of Invariant Measures," Journal of Theoretical Probability, Springer, vol. 16(3), pages 689-724, July.
    5. Saulius Minkevičius & Igor Katin & Joana Katina & Irina Vinogradova-Zinkevič, 2021. "On Little’s Formula in Multiphase Queues," Mathematics, MDPI, vol. 9(18), pages 1-15, September.
    6. Doruk Cetemen & Can Urgun & Leeat Yariv, 2023. "Collective Progress: Dynamics of Exit Waves," Journal of Political Economy, University of Chicago Press, vol. 131(9), pages 2402-2450.
    7. Zhang, Tonglin, 2024. "Variables selection using L0 penalty," Computational Statistics & Data Analysis, Elsevier, vol. 190(C).
    8. Alevizos, Filippos & Bagkavos, Dimitrios & Ioannides, Dimitrios, 2019. "Efficient estimation of a distribution function based on censored data," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 359-364.
    9. Cetemen, Doruk & Hwang, Ilwoo & Kaya, Ayça, 2020. "Uncertainty-driven cooperation," Theoretical Economics, Econometric Society, vol. 15(3), July.
    10. Ariane Hanebeck & Bernhard Klar, 2021. "Smooth distribution function estimation for lifetime distributions using Szasz–Mirakyan operators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(6), pages 1229-1247, December.
    11. Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani, 2019. "The De Vylder-Goovaerts conjecture holds true within the diffusion limit," Post-Print hal-01887402, HAL.
    12. Penrose, Mathew D., 2000. "Central limit theorems for k-nearest neighbour distances," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 295-320, February.
    13. Grégoire, Gérard & Hamrouni, Zouhir, 2002. "Change Point Estimation by Local Linear Smoothing," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 56-83, October.
    14. Scalas, Enrico & Viles, Noèlia, 2014. "A functional limit theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 385-410.
    15. Avishai Mandelbaum & Petar Momčilović, 2008. "Queues with Many Servers: The Virtual Waiting-Time Process in the QED Regime," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 561-586, August.
    16. Lloyd, Chris J. & Yong, Zhou, 1999. "Kernel estimators of the ROC curve are better than empirical," Statistics & Probability Letters, Elsevier, vol. 44(3), pages 221-228, September.
    17. Gromoll, H. Christian & Terwilliger, Bryce & Zwart, Bert, 2020. "Heavy traffic limit for the workload plateau process in a tandem queue with identical service times," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1435-1460.
    18. Tyran-Kaminska, Marta, 2010. "Convergence to Lévy stable processes under some weak dependence conditions," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1629-1650, August.
    19. Bouzebda, Salim & Slaoui, Yousri, 2022. "Nonparametric recursive method for moment generating function kernel-type estimators," Statistics & Probability Letters, Elsevier, vol. 184(C).
    20. Gianmarco Bet & Remco van der Hofstad & Johan S. H. van Leeuwaarden, 2019. "Heavy-Traffic Analysis Through Uniform Acceleration of Queues with Diminishing Populations," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 821-864, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:62:y:1996:i:2:p:327-345. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.