On the prospective minimum of the random walk conditioned to stay nonnegative
-
Vladimir A. Vatutin
and
Elena E. Dyakonova
Abstract
Let
be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants an that provide convergence as n → ∞ of the distributions of the sequence {Sn/an, n = 1, 2, …} to this stable law. Let Lr,n = minr≤m≤n Sm be the minimum of the random walk on the interval [r, n]. It is shown that
can have five different expressions, the forms of which depend on the relationships between the parameters r, k and n.
Originally published in Diskretnaya Matematika (2024) 36, №3, 50–79 (in Russian).
Funding statement: This work was supported by the Russian Science Foundation under grant no.24-11-00037 https://rscf.ru/en/project/24-11-00037/
Acknowledgment
In conclusion, we would like to thank an anonymous reviewer for the constructive comments that allow us to improve the presentation of results of the paper.
References
[1] Afanasyev V. I., Geiger J., Kersting G., Vatutin, V. A. “Criticality for branching processes in random environment”, Ann. Probab., 33:2 (2005), 645–673.10.1214/009117904000000928Search in Google Scholar
[2] Bertoin J., Doney R. A., “On conditioning a random walk to stay nonnegative”, Ann. Probab., 22:4 (1994), 2152–2167.10.1214/aop/1176988497Search in Google Scholar
[3] Bolthausen E., “On a functional central limit theorem for random walks conditioned to stay positive”, Ann. Probab., 4:3 (1976), 480–485.10.1214/aop/1176996098Search in Google Scholar
[4] Borovkov K.A., Vatutin V.A., “Reduced critical branching processes in random environment”, Stochastic Process. Appl., 71:2 (1997), 225–240.10.1016/S0304-4149(97)00074-4Search in Google Scholar
[5] Chaumont L., “Excursion normalisée, méandre et pont pour les processus de Lévy stables”, Bull. Sci. Math., 121 (1997), 377–403.Search in Google Scholar
[6] Caravenna F., “A local limit theorem for random walks conditioned to stay positive”, Probab.Theory Related Fields, 133 (2005), 508–530.10.1007/s00440-005-0444-5Search in Google Scholar
[7] Caravenna F., Chaumont L. Invariance principles for random walks conditioned to stay positive, Ann. Ins. H. Poincare Probab. Statist., 44 (2008), 170–190.10.1214/07-AIHP119Search in Google Scholar
[8] Caravenna F., Chaumont L., “An invariance principle for random walk bridges conditioned to stay positive”, Electron. J. Probab., 18:60 (2013), 1–32.10.1214/EJP.v18-2362Search in Google Scholar
[9] Chaumont L., Doney R. A., “Invariance principles for local times at the maximum of random walks and Levy processes”, Ann. Probab., 38 (2010), 1368–1389.10.1214/09-AOP512Search in Google Scholar
[10] Doney R.A., “Local behavior of first passage probabilities”, Probab. Theory Relat. Fields, 152:3-4 (2012), 559–588.10.1007/s00440-010-0330-7Search in Google Scholar
[11] Durrett R., “Conditioned limit theorems for some null recurrent Markov chains”, Ann. Probab., 6 (1978), 798–828.10.1214/aop/1176995430Search in Google Scholar
[12] Feller W., An Introduction to Probability Theory and Its Applications, 2, 2nd edition, Wiley, New York, 1971.Search in Google Scholar
[13] Gnedenko B. V., Kolmogorov A. N., Limit distributions for sums of independent random variables, Addison-Wesley, 1954.Search in Google Scholar
[14] Iglehart D. L., “Functional central limit theorems for random walks conditioned to stay positive”, Ann. Probab., 2 (1974), 608–619.10.1214/aop/1176996607Search in Google Scholar
[15] Ito K., McKean H. P. Jr., Diffusion Processes and their Sample Paths, Springer Science & Business Media, 2012, 323 pp.Search in Google Scholar
[16] Rogozin B.A., “On the distrbution of the first ladder moment and height and fluctuations of a random walk”, Theory Probab. Appl., 16 (1971), 575–595.10.1137/1116067Search in Google Scholar
[17] Sinai Ya. G., “On the distribution of the first positive sum for a sequence of independent random variables”, Theory Probab. Appl., 2 (1957), 122–129.10.1137/1102009Search in Google Scholar
[18] Urbe Bravo G., “Bridges of Levy processes conditioned to stay positive”, Bernoulli, 20:1 (2014), 190–206.10.3150/12-BEJ481Search in Google Scholar
[19] Vatutin V. A., “Reduced branching processes in random environment: the critical case”, Theory Probab. Appl., 47:1 (2003), 99–113.10.1137/S0040585X97979421Search in Google Scholar
[20] Vatutin V., Dyakonova E., “Critical branching processes evolving in an unfavorable random environment”, Discrete Math. Appl., 34:3 (2024), 175–186.Search in Google Scholar
[21] Vatutin V. A., Dong G., Dyakonova E. E., “Random walks conditioned to stay non-negative and branching processes in non-favorable random environment”, Sb. Math., 214:11 (2023), 1501–1533.10.4213/sm9908eSearch in Google Scholar
[22] Vatutin V.A., Wachtel V., “Local probabilities for random walks conditioned to stay positive”, Probab. Theory Related Fields, 143 (2009), 177–217.10.1007/s00440-007-0124-8Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On continuants of continued fractions with rational partial quotients
- On linear equivalence of piecewise-linear permutations of the field 𝔽2n
- On the prospective minimum of the random walk conditioned to stay nonnegative
- Limiting behavior of percolation cluster in a multilayered random environment with breakdown
Articles in the same Issue
- Frontmatter
- On continuants of continued fractions with rational partial quotients
- On linear equivalence of piecewise-linear permutations of the field 𝔽2n
- On the prospective minimum of the random walk conditioned to stay nonnegative
- Limiting behavior of percolation cluster in a multilayered random environment with breakdown
