Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T04:00:54.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Implicit renewal theory in the arithmetic case

Part of: Special processes Stochastic analysis

Published online by Cambridge University Press:  15 September 2017

Péter Kevei
Péter Kevei*
Affiliation:
Technische Universität München and MTA–SZTE Analysis and Stochastics Research Group
*
* Postal address: University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution X of the fixed point equations X =DAX + B and X =DAX ∨ B is ℓ(x) q(x) x-κ, where q is a logarithmically periodic function q(x eh) = q(x), x > 0, with h being the span of the arithmetic distribution of log A, and ℓ is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevičius (1975) and Goldie (1991).

Keywords

Perpetuity equationmaximum of perturbed random walkimplicit renewal theoremarithmetic distributioniterated function system

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 732 - 749
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alsmeyer, G. (2016). On the stationary tail index of iterated random Lipschitz functions. Stoch. Process. Appl. 126, 209–233. CrossRefGoogle Scholar
[2] Alsmeyer, G., Biggins, J. D. and Meiners, M. (2012). The functional equation of the smoothing transform. Ann. Prob. 40, 2069–2105. CrossRefGoogle Scholar
[3] Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. (New York) 51), 2nd edn. Springer, New York. Google Scholar
[4] Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Prob. 16, 489–518. CrossRefGoogle Scholar
[5] Berkes, I., Györfi, L. and Kevei, P. (2017). Tail probabilities of St. Petersburg sums, trimmed sums, and their limit. J. Theoret. Prob. 30, 1104–1129. CrossRefGoogle Scholar
[6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press. Google Scholar
[7] Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails: The Equation X=AX+B. Springer, Cham. CrossRefGoogle Scholar
[8] Caravenna, F. and Doney, R. (2016). Local large deviations and the strong renewal theorem. Preprint. Available at https://arxiv.org/abs/1612.07635. Google Scholar
[9] Csörgő, S. (2002). Rates of merge in generalized St. Petersburg games. Acta Sci. Math. (Szeged) 68, 815–847. Google Scholar
[10] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45–76. CrossRefGoogle Scholar
[11] Durrett, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275–301. CrossRefGoogle Scholar
[12] Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263–291. CrossRefGoogle Scholar
[13] Foss, S., Korshunov, D. and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York. CrossRefGoogle Scholar
[14] Garsia, A. and Lamperti, J. (1962/1963). A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37, 221–234. CrossRefGoogle Scholar
[15] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166. CrossRefGoogle Scholar
[16] Grincevičjus, A. K. (1975). One limit distribution for a random walk on lines. Lithuanian Math. J. 15, 580–589. CrossRefGoogle Scholar
[17] Iksanov, A. (2007). Perpetuities, Branching Random Walk and Selfdecomposability. KTI-PRINT, Kiev (in Ukranian). Google Scholar
[18] Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Birkhäuser, Cham. CrossRefGoogle Scholar
[19] Jelenković, P. R. and Olvera-Cravioto, M. (2012). Implicit renewal theorem for trees with general weights. Stoch. Process Appl. 122, 3209–3238. CrossRefGoogle Scholar
[20] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248. CrossRefGoogle Scholar
[21] Kesten, H. and Maller, R. A. (1996). Two renewal theorems for general random walks tending to infinity. Prob. Theory Relat. Fields 106, 1–38. CrossRefGoogle Scholar
[22] Kevei, P. (2016). A note on the Kesten–Grincevičius–Goldie theorem. Electron. Commun. Prob. 21, 51. CrossRefGoogle Scholar
[23] Korshunov, D. A. (2005). The critical case of the Cramér-Lundberg theorem on the asymptotics tail behaviour of the maximum of a negative drift random walk. Siberian Math. J. 46, 1077–1081. CrossRefGoogle Scholar
[24] Lau, K.-S. and Rao, C. R. (1982). Integrated Cauchy functional equation and characterizations of the exponential law. Sankhyā A 44, 72–90, 452. Google Scholar
[25] Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York. Google Scholar
[26] Megyesi, Z. (2000). A probabilistic approach to semistable laws and their domains of partial attraction. Acta Sci. Math. (Szeged) 66, 403–434. Google Scholar
[27] Megyesi, Z. (2002). Domains of geometric partial attraction of max-semistable laws: structure, merge and almost sure limit theorems. J. Theoret. Prob. 15, 973–1005. CrossRefGoogle Scholar