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

  • Valeriy I. Afanasyev EMAIL logo
Veröffentlicht/Copyright: 7. April 2021
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Discrete Mathematics and Applications
Aus der Zeitschrift Discrete Mathematics and Applications Band 31 Heft 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.10.1007/978-1-4757-4229-9Suche in Google Scholar

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[3] Billingsley P., Convergence of Probability Measures, New York: John Wiley & Sons, 1968.Suche 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-2019-0014Suche in Google Scholar

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

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
random walks with zero drift; boundary problems; limit theorems

Artikel in diesem Heft

  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
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