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

Part of: Distribution theory Distribution theory - Probability Qualitative behavior

Published online by Cambridge University Press:  26 July 2018

Dmitri Finkelshtein  and
Pasha Tkachov
Dmitri Finkelshtein*
Affiliation:
Swansea University
Pasha Tkachov*
Affiliation:
Universität Bielefeld
*
* Postal address: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK. Email address: d.l.finkelshtein@swansea.ac.uk
** Current address: Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 L'Aquila AQ, Italy. Email address: pasha.tkachov@gssi.it
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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.

Keywords

Subexponential densitylong-tail functionconvolution tailheavy-tailed distributiontail-equivalenceasymptotic behavior

MSC classification

Primary: 60E05: Distributions
Secondary: 45M05: Asymptotics 62E20: Asymptotic distribution theory
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 373 - 395
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. Google Scholar
[2]Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Prob. 16, 489–518. Google Scholar
[3]Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. Google Scholar
[4]Borovkov, A. A. and Borovkov, K. A. (2008). Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions. Cambridge University Press. Google Scholar
[5]Brändle, C., Chasseigne, E. and Ferreira, R. (2011). Unbounded solutions of the nonlocal heat equation. Commun. Pure Appl. Anal. 10, 1663–1686. Google Scholar
[6]Chasseigne, E., Chaves, M. and Rossi, J. D. (2006). Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 986, 271–291. Google 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, 640–648. Google Scholar
[8]Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Analyse Math. 26, 255–302. Google Scholar
[9]Cline, D. B. H. and Resnick, S. I. (1992). Multivariate subexponential distributions. Stoch. Process. Appl. 42, 49–72. Google Scholar
[10]Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer, Berlin. Google 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. Google 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, 366–383. Google Scholar
[14]Foss, S., Korshunov, D. and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York. Google Scholar
[15]Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269. Google Scholar
[16]Kondratiev, Y., Molchanov, S., Piatnitski, A. and Zhizhina, E. (2018). Resolvent bounds for jump generators. Applicable Anal. 97, 323–336. Google Scholar
[17]Omey, E. A. M. (2006). Subexponential distribution functions in ℝd. J. Math. Sci. (N.Y.) 138, 5434–5449. Google Scholar
[18]Rogozin, B. A. and Sgibnev, M. S. (1999). Strongly subexponential distributions, and Banach algebras of measures. Siberian Math. J. 40, 963–971. Google Scholar
[19]Samorodnitsky, G. and Sun, J. (2016). Multivariate subexponential distributions and their applications. Extremes 19, 171–196. Google Scholar
[20]Sgibnev, M. S. (1981). Banach algebras of functions with the same asymptotic behavior at infinity. Siberian Math. J. 22, 467–473. Google Scholar
[21]Sgibnev, M. S. (1990). Asymptotics of infinitely divisible distributions on ℝ. Siberian Math. J. 31, 115–119. Google Scholar
[22]Watanabe, T. (2008). Convolution equivalence and distributions of random sums. Prob. Theory Relat. Fields 142, 367–397. Google Scholar