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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

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

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:

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
\Bibitem{BorBor01} \by A.~A.~Borovkov, K.~A.~Borovkov \paper On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails \jour Teor. Veroyatnost. i Primenen. \yr 2001 \vol 46 \issue 2 \pages 209--232 \mathnet{http://mi.mathnet.ru/tvp3915} \crossref{https://doi.org/10.4213/tvp3915} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1968683} \zmath{https://zbmath.org/?q=an:1006.60020} \transl \jour Theory Probab. Appl. \yr 2002 \vol 46 \issue 2 \pages 193--213 \crossref{https://doi.org/10.1137/S0040585X97978877} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000176400600001} 

• http://mi.mathnet.ru/eng/tvp3915
• https://doi.org/10.4213/tvp3915
• http://mi.mathnet.ru/eng/tvp/v46/i2/p209

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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
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
3. A. A. Borovkov, “Kolmogorov and boundary problems of probability theory”, Russian Math. Surveys, 59:1 (2004), 91–102
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
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
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
7. A. A. Borovkov, “Asymptotic analysis for random walks with nonidentically distributed jumps having finite variance”, Siberian Math. J., 46:6 (2005), 1020–1038
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
9. Barbe P., Mccormick W.P., “Asymptotic expansions of convolutions of regularly varying distributions”, J. Aust. Math. Soc., 78:3 (2005), 339–371
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
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
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
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
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
15. Blanchet J.H., Liu Jingchen, “State-dependent importance sampling for regularly varying random walks”, Adv. in Appl. Probab., 40:4 (2008), 1104–1128
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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
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
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
20. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868
21. Mao T., Hu T., “Second-Order Properties of Risk Concentrations Without the Condition of Asymptotic Smoothness”, Extremes, 16:4 (2013), 383–405
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
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