Home Two-sided problem for the random walk with bounded maximal increment
Article
Licensed
Unlicensed Requires Authentication

Two-sided problem for the random walk with bounded maximal increment

  • Valeriy I. Afanasyev EMAIL logo
Published/Copyright: April 7, 2021
Become an author with De Gruyter Brill
Author Information Explore this Subject
Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 31 Issue 2

Abstract

We consider a random walk with zero drift and finite positive variance σ2. For positive numbers y, z we find the limit as n→∞ of the probability that the first exit of the walk from interval -zσn,yσn occurs through its left end, while the maximum increment of the walk until the exit is smaller than xσn, where x is a positive number. The limit theorem is established for the moment of the first exit of the walk from the indicated interval under the condition that this exit occurs through its left end and the value of the maximum walk increment is bounded.

Keywords: random walks with zero drift; boundary problems; limit theorems

Note: Originally published in Diskretnaya Matematika (2019) 31, No_3, 3–16 (in Russian).

References

[1] Spitzer F., Principles of Random Walk, Springer-Verlag New York, 1964, XIII, 408 pp.10.1007/978-1-4757-4229-9Search in Google Scholar

[2] Afanasyev V. I., “On the times of attaining high levels by a random walk in a random environment”, Theory Probab. Appl., 65:3, 359–374.10.1137/S0040585X97T990009Search in Google Scholar

[3] Billingsley P., Convergence of Probability Measures, New York: John Wiley & Sons, 1968.Search in Google Scholar

[4] Afanasyev V. I., “Convergence to the local time of Brownian meander”, Discrete Math. Appl., 29:3 (2019), 149–158.10.1515/dma-2019-0014Search in Google Scholar

[5] Shimura M., “A class of conditional limit theorems related to ruin problem”, Ann. Probab., 11:1 (1983), 40–45.10.1214/aop/1176993658Search in Google Scholar

[6] Doney R.A., “A note on conditioned random walk”, J. Appl. Probab., 20:2 (1983), 409–412.10.2307/3213815Search in Google Scholar

[7] Feller W., An Introduction to Probability Theory and Its Applications. V. 2, 2nd ed., John Wiley & Sons, 1971, 669 pp.Search in Google Scholar

Received: 2019-03-03
Published Online: 2021-04-07
Published in Print: 2021-04-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Two-sided problem for the random walk with bounded maximal increment
  3. On the use of binary operations for the construction of a multiply transitive class of block transformations
  4. On the complexity of implementation of a system of two monomials by composition circuits
  5. On the degree of restrictions of q-valued logic vector functions to linear manifolds
  6. Trees with a given number of leaves and the maximal number of maximum independent sets
  7. Size distribution of the largest component of a random A-mapping
https://doi.org/10.1515/dma-2021-0008
Keywords for this article
random walks with zero drift; boundary problems; limit theorems

Articles in the same Issue

  1. Frontmatter
  2. Two-sided problem for the random walk with bounded maximal increment
  3. On the use of binary operations for the construction of a multiply transitive class of block transformations
  4. On the complexity of implementation of a system of two monomials by composition circuits
  5. On the degree of restrictions of q-valued logic vector functions to linear manifolds
  6. Trees with a given number of leaves and the maximal number of maximum independent sets
  7. Size distribution of the largest component of a random A-mapping
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 24.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dma-2021-0008/html
Scroll to top button