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Kesten's bound for subexponential densities on the real line and its multi-dimensional analogues

Published online by Cambridge University Press:  26 July 2018

Dmitri Finkelshtein
Affiliation:
Swansea University
Pasha Tkachov
Affiliation:
Universität Bielefeld
Corresponding

Abstract

We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is asymptotically equivalent to this density multiplied by n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.

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Original Article
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Andreu-Vaillo, F., Mazón, J. M., Rossi, J. D. and Toledo-Melero, J. J. (2010). Nonlocal Diffusion Problems. American Mathematical Society, Providence, RI. CrossRefGoogle Scholar
[2]Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Prob. 16, 489518. CrossRefGoogle Scholar
[3]Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. CrossRefGoogle Scholar
[4]Borovkov, A. A. and Borovkov, K. A. (2008). Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions. Cambridge University Press. CrossRefGoogle Scholar
[5]Brändle, C., Chasseigne, E. and Ferreira, R. (2011). Unbounded solutions of the nonlocal heat equation. Commun. Pure Appl. Anal. 10, 16631686. CrossRefGoogle Scholar
[6]Chasseigne, E., Chaves, M. and Rossi, J. D. (2006). Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 986, 271291. CrossRefGoogle Scholar
[7]Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Prob. Appl. 9, 640648. CrossRefGoogle Scholar
[8]Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Analyse Math. 26, 255302. CrossRefGoogle Scholar
[9]Cline, D. B. H. and Resnick, S. I. (1992). Multivariate subexponential distributions. Stoch. Process. Appl. 42, 4972. CrossRefGoogle Scholar
[10]Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer, Berlin. CrossRefGoogle Scholar
[11]Finkelshtein, D. and Tkachov, P. (2017). Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line. Applicable Anal. 10.1080/00036811.2017.1400537. CrossRefGoogle Scholar
[12]Finkelshtein, D., Kondratiev, Y. and Tkachov, P. (2017). Accelerated front propagation for monostable equations with nonlocal diffusion: multidimensional case. Preprint. Available at https://arxiv.org/abs/1611.09329. Google Scholar
[13]Foss, S. and Korshunov, D. (2007). Lower limits and equivalences for convolution tails. Ann. Prob. 35, 366383. CrossRefGoogle Scholar
[14]Foss, S., Korshunov, D. and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York. CrossRefGoogle Scholar
[15]Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259269. CrossRefGoogle Scholar
[16]Kondratiev, Y., Molchanov, S., Piatnitski, A. and Zhizhina, E. (2018). Resolvent bounds for jump generators. Applicable Anal. 97, 323336. CrossRefGoogle Scholar
[17]Omey, E. A. M. (2006). Subexponential distribution functions in ℝd. J. Math. Sci. (N.Y.) 138, 54345449. CrossRefGoogle Scholar
[18]Rogozin, B. A. and Sgibnev, M. S. (1999). Strongly subexponential distributions, and Banach algebras of measures. Siberian Math. J. 40, 963971. CrossRefGoogle Scholar
[19]Samorodnitsky, G. and Sun, J. (2016). Multivariate subexponential distributions and their applications. Extremes 19, 171196. CrossRefGoogle Scholar
[20]Sgibnev, M. S. (1981). Banach algebras of functions with the same asymptotic behavior at infinity. Siberian Math. J. 22, 467473. CrossRefGoogle Scholar
[21]Sgibnev, M. S. (1990). Asymptotics of infinitely divisible distributions on ℝ. Siberian Math. J. 31, 115119. CrossRefGoogle Scholar
[22]Watanabe, T. (2008). Convolution equivalence and distributions of random sums. Prob. Theory Relat. Fields 142, 367397. CrossRefGoogle Scholar

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