The Self-Similar Dynamics of Renewal Processes

Albert Meads Fisher (University of Sao Paulo)
Marina Talet (Université de Provence)


We prove an almost sure invariance principle in log density for renewal processes with gaps in the domain of attraction of an $\alpha$-stable law. There are three different types of behavior: attraction to a Mittag-Leffler process for $0<\alpha<1$, to a centered Cauchy process for $\alpha=1$ and to a stable process for $1<\alpha\leq 2$. Equivalently, in dynamical terms, almost every renewal path is, upon centering and up to a regularly varying coordinate change of order one, and after removing a set of times of Cesàro density zero, in the stable manifold of a self-similar path for the scaling flow. As a corollary we have pathwise functional and central limit theorems.

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Pages: 929-961

Publication Date: May 10, 2011

DOI: 10.1214/EJP.v16-888


  1. Berkes, I.; Dehling, H. Some limit theorems in log density. Ann. Probab. 21 (1993), no. 3, 1640--1670. MR1235433 (94h:60026)
  2. Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2 MR0898871 (88i:26004)
  3. Billingsley, P. . Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396 (38 #1718)
  4. Bingham, N. H. Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 1971 1--22. MR0281255 (43 #6974)
  5. Breiman, L.. On the tail behavior of sums of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 1967 20--25. MR0226707 (37 #2294)
  6. Cs?rgő, M.; RÈvÈsz, P. Strong approximations in probability and statistics. Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. 284 pp. ISBN: 0-12-198540-7 MR0666546 (84d:60050)
  7. de Acosta, A.; GinÈ, E.. Convergence of moments and related functionals in the general central limit theorem in Banach spaces. Z. Wahrsch. Verw. Gebiete 48 (1979), no. 2, 213--231. MR0534846 (80h:60011)
  8. Darling, D. A.; Kac, M. On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84 (1957), 444--458. MR0084222 (18,832a)
  9. Durrett, R.. Probability. Theory and examples. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. x+453 pp. ISBN: 0-534-13206-5 MR1068527 (91m:60002)
  10. Feller, W.. Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, (1949). 98--119. MR0032114 (11,255c)
  11. Fisher, A. M.. A Pathwise Central Limit Theorem for random walks. Univ. of Goettingen preprint series, (1987); accepted for publication in the Annals of Probability (1989).
  12. Fisher, A.M. . Convex-invariant means and a pathwise central limit theorem. Adv. in Math. 63 (1987), no. 3, 213--246. MR0877784 (88g:60058)
  13. Fisher, A.M.; Lopes, A.; Talet, M. Self-similar returns in the transition from finite to infinite measure. In preparation.
  14. Fomin, S. Finite invariant measures in the flows. (Russian) Rec. Math. [Mat. Sbornik] N. S. 12(54), (1943). 99--108. MR0009097 (5,101b)
  15. Fisher, A.M.; Talet, M. Dynamical attraction to stable processes. To appear in Annales de l'Institut Henri Poincare.
  16. Fisher, A.M.; Talet, M. Log averages and self-similar return sets. Submitted.
  17. Horv·th, L.. Strong approximation of renewal processes. Stochastic Process. Appl. 18 (1984), no. 1, 127--138. MR0757352 (85k:60046)
  18. KomlÛs, J.; Major, P.; Tusn·dy, G. An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), no. 1, 33--58. MR0402883 (53 #6697)
  19. Major, P.. Approximation of partial sums of i.i.d.r.v.s. when the summands have only two moments. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 3, 221--229. MR0415744 (54 #3824)
  20. Major, P.. The approximation of partial sums of independent RV's. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 3, 213--220. MR0415743 (54 #3823)
  21. Skorohod, A. V. Limit theorems for stochastic processes with independent increments. (Russian) Teor. Veroyatnost. i Primenen. 2 1957 145--177. MR0094842 (20 #1351)
  22. Stone, C.. Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc. 14 1963 694--696. MR0153046 (27 #3015)
  23. Strassen, V. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 1964 211--226 (1964). MR0175194 (30 #5379)
  24. Strassen, V.. Almost sure behavior of sums of independent random variables and martingales. 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Vol. II: Contributions to Probability Theory, Part 1, pp. 315--343 Univ. California Press, Berkeley, Calif. MR0214118 (35 #4969)
  25. Vervaat, W.. Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 245--253. MR0321164 (47 #9697)
  26. Walters, P.. An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. ix+250 pp. ISBN: 0-387-90599-5 MR0648108 (84e:28017)
  27. Whitt, W.. Some useful functions for functional limit theorems. Math. Oper. Res. 5 (1980), no. 1, 67--85. MR0561155 (81e:60035)
  28. Whitt, W.. Stochastic-process limits. An introduction to stochastic-process limits and their application to queues. Springer Series in Operations Research. Springer-Verlag, New York, 2002. xxiv+602 pp. ISBN: 0-387-95358-2 MR1876437 (2003f:60005)

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