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Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 2, Pages 209–232 (Mi tvp3915)  

This article is cited in 22 scientific papers (total in 23 papers)

On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails

A. A. Borovkova, K. A. Borovkovb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b University of Melbourne, Department of Mathematics and Statistics

Abstract: We establish first-order approximations and asymptotic expansions for probabilities of crossing arbitrary curvilinear boundaries in the large deviations range by random walks with regularly varying distribution tails. In particular, we study the large deviations probabilities for the sums and maxima of partial sums of independent and identically distributed random variables, including the asymptotic behavior of the densities when they exist. Extensions to the "regular exponential" case (when the distribution tail differs from the exponential one by a regularly varying factor) are considered in part II of the paper.

Keywords: large deviations, random walk, regular variation.

DOI: https://doi.org/10.4213/tvp3915

Full text: PDF file (2155 kB)

English version:
Theory of Probability and its Applications, 2002, 46:2, 193–213

Bibliographic databases:

Received: 23.05.2000

Citation: A. A. Borovkov, K. A. Borovkov, “On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails”, Teor. Veroyatnost. i Primenen., 46:2 (2001), 209–232; Theory Probab. Appl., 46:2 (2002), 193–213

Citation in format AMSBIB
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\jour Theory Probab. Appl.
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    This publication is cited in the following articles:
    1. Borovkov A.A., “Large deviations probabilities for random walks in the absence of finite expectations of jumps”, Probab. Theory Related Fields, 125:3 (2003), 421–446  crossref  mathscinet  zmath  isi  scopus
    2. Foss S., Zachary S., “The maximum on a random time interval of a random walk with long-tailed increments and negative drift”, Ann. Appl. Probab., 13:1 (2003), 37–53  crossref  mathscinet  zmath  isi  scopus
    3. A. A. Borovkov, “Kolmogorov and boundary problems of probability theory”, Russian Math. Surveys, 59:1 (2004), 91–102  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. A. Borovkov, K. A. Borovkov, “On probabilities of large deviations for random walks. II. Regular exponentially decaying distributions”, Theory Probab. Appl., 49:3 (2005), 189–206  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Tang Qihe, “Uniform estimates for the tail probability of maxima over finite horizons with subexponential tails”, Probab. Engrg. Inform. Sci., 18:1 (2004), 71–86  crossref  mathscinet  zmath  isi  scopus
    6. A. A. Borovkov, K. A. Borovkov, “Large Deviations Probabilities for Generalized Renewal Processes with Regularly Varying Jump Distributions”, Siberian Adv. Math., 16:1 (2006), 1–65  mathnet  mathscinet
    7. A. A. Borovkov, “Asymptotic analysis for random walks with nonidentically distributed jumps having finite variance”, Siberian Math. J., 46:6 (2005), 1020–1038  mathnet  crossref  mathscinet  zmath  isi  elib
    8. Foss S., Palmowski Z., Zachary S., “The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk”, Ann. Appl. Probab., 15:3 (2005), 1936–1957  crossref  mathscinet  zmath  isi  scopus
    9. Barbe P., Mccormick W.P., “Asymptotic expansions of convolutions of regularly varying distributions”, J. Aust. Math. Soc., 78:3 (2005), 339–371  crossref  mathscinet  zmath  isi  elib
    10. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II”, Theory Probab. Appl., 51:4 (2007), 567–594  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I”, Theory Probab. Appl., 51:2 (2007), 227–255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. A. A. Borovkov, A. A. Mogul'skii, L. V. Rozovskii, A. I. Sakhanenko, “On Zhulev's paper “On large deviations. II””, Theory Probab. Appl., 51:2 (2007), 398–400  mathnet  crossref  crossref  mathscinet  zmath  elib
    13. A. A. Borovkov, A. A. Mogul'skii, “On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. I”, Theory Probab. Appl., 53:2 (2009), 301–311  mathnet  crossref  crossref  zmath  isi
    14. A. A. Borovkov, A. A. Mogul'skii, “On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. II”, Theory Probab. Appl., 53:4 (2009), 573–593  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. Blanchet J.H., Liu Jingchen, “State-dependent importance sampling for regularly varying random walks”, Adv. in Appl. Probab., 40:4 (2008), 1104–1128  crossref  mathscinet  zmath  isi  scopus
    16. Barbe Ph., McCormick W.P., “Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications”, Mem. Amer. Math. Soc., 197, no. 922, 2009, viii+117 pp.  mathscinet  isi
    17. Kortschak D., Albrecher H., “An asymptotic expansion for the tail of compound sums of Burr distributed random variables”, Stat. Probab. Lett., 80:7-8 (2010), 612–620  crossref  mathscinet  zmath  isi  elib  scopus
    18. Fushiya H., Kusuoka Sh., “Uniform Estimate for Distributions of the Sum of i.i.d. Random Variables with Fat Tail”, Journal of Mathematical Sciences-the University of Tokyo, 17:1 (2010), 79–121  mathscinet  zmath  isi
    19. Albrecher H., Hipp Ch., Kortschak D., “Higher-order expansions for compound distributions and ruin probabilities with subexponential claims”, Scandinavian Actuarial Journal, 2010, no. 2, 105–135  crossref  mathscinet  zmath  isi  elib  scopus
    20. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. Mao T., Hu T., “Second-Order Properties of Risk Concentrations Without the Condition of Asymptotic Smoothness”, Extremes, 16:4 (2013), 383–405  crossref  mathscinet  zmath  isi  scopus
    22. Lin J., “Second Order Tail Behaviour For Heavy-Tailed Sums and Their Maxima With Applications To Ruin Theory”, Extremes, 17:2 (2014), 247–262  crossref  mathscinet  zmath  isi  scopus
    23. Blanchet J. Murthy K.R.A., “Tail Asymptotics For Delay in a Half-Loaded Gi/Gi/2 Queue With Heavy-Tailed Job Sizes”, Queueing Syst., 81:4 (2015), 301–340  crossref  mathscinet  zmath  isi  elib  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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