The Annals of Probability

Recurrence rates and hitting-time distributions for random walks on the line

Françoise Pène, Benoît Saussol, and Roland Zweimüller

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We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as $\varepsilon\to0$, of the return time to $(-\varepsilon,\varepsilon)$. We then refine this result by establishing a limit theorem for the hitting-time distributions of $(x-\varepsilon,x+\varepsilon)$ with arbitrary $x\in\mathbb{R} $.

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Ann. Probab., Volume 41, Number 2 (2013), 619-635.

First available in Project Euclid: 8 March 2013

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems

Random walk stable distribution recurrence quantitative recurrence hitting time


Pène, Françoise; Saussol, Benoît; Zweimüller, Roland. Recurrence rates and hitting-time distributions for random walks on the line. Ann. Probab. 41 (2013), no. 2, 619--635. doi:10.1214/11-AOP698.

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  • [1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [2] Bressaud, X. and Zweimüller, R. (2001). Non exponential law of entrance times in asymptotically rare events for intermittent maps with infinite invariant measure. Ann. Henri Poincaré 2 501–512.
  • [3] Bretagnolle, J. and Dacunha-Castelle, D. (1968). Théorèmes limites à distance finie pour les marches aléatoires. Ann. Inst. H. Poincaré Sect. B (N.S.) 4 25–73.
  • [4] Cheliotis, D. (2006). A note on recurrent random walks. Statist. Probab. Lett. 76 1025–1031.
  • [5] Chung, K.-L. and Erdös, P. (1947). On the lower limit of sums of independent random variables. Ann. of Math. (2) 48 1003–1013.
  • [6] Dvoretzky, A. and Erdös, P. (1951). Some problems on random walk in space. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 353–367. Univ. California Press, Berkeley.
  • [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [8] Pène, F. and Saussol, B. (2009). Quantitative recurrence in two-dimensional extended processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 1065–1084.
  • [9] Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Statist. 36 546–551.