Asymptotic results for the absorption times of random walks with a barrier
Author:
Pavlo Negadaĭlov
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal:
Theor. Probability and Math. Statist. 79 (2009), 127-138
MSC (2000):
Primary 60J80, 60E99; Secondary 60G42
DOI:
https://doi.org/10.1090/S0094-9000-09-00785-6
Published electronically:
December 30, 2009
MathSciNet review:
2494542
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A sequence ,
,
, is called a random walk with a barrier
, where the
are independent copies of a random variable
assuming positive integer values. The asymptotic behavior of the absorption times is studied in the paper for a random walk with a barrier. This behavior depends on the properties of the tail of the distribution of the random variable
.
- 1. N. H. Bingham, Limit theorems for regenerative phenomena, recurrent events and renewal theory, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 (1972), 20–44. MR 353459, https://doi.org/10.1007/BF00535105
- 2. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1989. MR 1015093
- 3.
M. Drmota, A. Iksanov, M. Möhle, and U. Rösler, A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree, http://www.unicyb.kiev.ua/
iksan (2006).
- 4. William Feller, Fluctuation theory of recurrent events, Trans. Amer. Math. Soc. 67 (1949), 98–119. MR 32114, https://doi.org/10.1090/S0002-9947-1949-0032114-7
- 5. C. C. Heyde, A limit theorem for random walks with drift, J. Appl. Probability 4 (1967), 144–150. MR 207061, https://doi.org/10.2307/3212307
- 6. Karl Hinderer and Harro Walk, Anwendung von Erneuerungstheoremen und Taubersätzen für eine Verallgemeinerung der Erneuerungsprozesse, Math. Z. 126 (1972), 95–115 (German). MR 300354, https://doi.org/10.1007/BF01122317
- 7. K. Bruce Erickson, Strong renewal theorems with infinite mean, Trans. Amer. Math. Soc. 151 (1970), 263–291. MR 268976, https://doi.org/10.1090/S0002-9947-1970-0268976-9
- 8.
A. Iksanov and M. Möhle, On a random recursion related to absorption times of death Markov chains, http://www.unicyb.kiev.ua/
iksan (2007).
- 9. Alex Iksanov and Martin Möhle, A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree, Electron. Comm. Probab. 12 (2007), 28–35. MR 2407414, https://doi.org/10.1214/ECP.v12-1253
- 10. A. Meir and J. W. Moon, Cutting down recursive trees, Math. Biosci. 21 (1974), 173-181.
- 11. Wim Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab. 11 (1979), no. 4, 750–783. MR 544194, https://doi.org/10.2307/1426858
- 12. William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
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Additional Information
Pavlo Negadaĭlov
Affiliation:
Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
npasha@ukr.net
DOI:
https://doi.org/10.1090/S0094-9000-09-00785-6
Keywords:
Random walks with a barrier,
absorption moments
Received by editor(s):
October 18, 2007
Published electronically:
December 30, 2009
Additional Notes:
The research is supported by DFG, project 436UKR 113/93/0-1
Article copyright:
© Copyright 2009
American Mathematical Society