A transience condition for a class of one-dimensional symmetric Lévy processes

Nikola Sandrić (University of Zagreb)


In this paper, we give a sufficient condition for the transience for a class of one dimensional symmetric Lévy processes. More precisely, we prove that  a one dimensional symmetric Lévy process with the Lévy measure $\nu(dy)=f(y)dy$ or $\nu(\{n\})=p_n$, where the density function $f(y)$ is such that $f(y)>0$  a.e. and the sequence $\{p_n\}_{n\geq1}$ is such that $p_n>0$ for all $n\geq1$, is transient if $$\int_1^{\infty}\frac{dy}{y^{3}f(y)}<\infty\quad\mbox{or}\quad\sum_{n=1}^{\infty}\frac{1}{n^{3}p_n}<\infty.$$  Similarly, we derive an an alogous transience condition for one-dimensional symmetric random walks with continuous and discrete jumps.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-13

Publication Date: August 24, 2013

DOI: 10.1214/ECP.v18-2802


  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
  • Hobert, James P.; Schweinsberg, Jason. Conditions for recurrence and transience of a Markov chain on $\Bbb Z^ +$ and estimation of a geometric success probability. Ann. Statist. 30 (2002), no. 4, 1214--1223. MR1926175
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
  • Lyons, Terry. A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11 (1983), no. 2, 393--402. MR0690136
  • Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6 MR1287609
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4 MR1739520
  • Shepp, L. A. Symmetric random walk. Trans. Amer. Math. Soc. 104 1962 144--153. MR0139212
  • Shepp, L. A. Recurrent random walks with arbitrarily large steps. Bull. Amer. Math. Soc. 70 1964 540--542. MR0169305

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.