Quenched small deviation for the trajectory of a random walk with random environment in time. (English) Zbl 1533.60188
Theory Probab. Appl. 68, No. 2, 267-284 (2023) and Teor. Veroyatn. Primen. 68, No. 2, 322-343 (2023).
Summary: We consider the small deviation probability for a random walk with random environment in time. Compared to [A. A. Mogul’skij, Theory Probab. Appl. 19, 726–736 (1974; Zbl 0326.60061); translation from Teor. Veroyatn. Primen. 19, 755–765 (1974)], for the independent and identically distributed (i.i.d.) random walk, the rate is smaller (due to the random environment), which is specified in terms of the quenched and annealed variance.
MSC:
| 60K37 | Processes in random environments |
| 60G50 | Sums of independent random variables; random walks |
| 60F10 | Large deviations |
Keywords:
random environment; small deviation probability; partial sums of independent random variablesCitations:
Zbl 0326.60061References:
| [1] | A. A. Mogul’skii, Small deviations in the space of trajectories, Theory Probab. Appl., 19 (1975), pp. 726-736, https://doi.org/10.1137/1119081. · Zbl 0326.60061 |
| [2] | A. de Acosta, Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann. Probab., 11 (1983), pp. 78-101, https://doi.org/10.1214/aop/1176993661. · Zbl 0504.60033 |
| [3] | M. Talagrand, New Gaussian estimates for enlarged balls, Geom. Funct. Anal., 3 (1993), pp. 502-526, https://doi.org/10.1007/BF01896240. · Zbl 0815.46021 |
| [4] | Y. Xiao, Hausdorff-type measures of the sample paths of fractional Brownian motion, Stochastic Process. Appl., 74 (1998), pp. 251-272, https://doi.org/10.1016/S0304-4149(97)00119-1. · Zbl 0937.60076 |
| [5] | N. Gantert, Y. Hu, and Z. Shi, Asymptotics for the survival probability in a killed branching random walk, Ann. Inst. H. Poincaré Probab. Statist., 47 (2011), pp. 111-129, https://doi.org/10.1214/10-AIHP362. · Zbl 1210.60093 |
| [6] | B. Jaffuel, The critical barrier for the survival of branching random walk with absorption, Ann. Inst. H. Poincaré Probab. Statist., 48 (2012), pp. 989-1009, https://doi.org/10.1214/11-AIHP453. · Zbl 1263.60076 |
| [7] | J. Kuelbs and W. V. Li, Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal., 116 (1993), pp. 133-157, https://doi.org/10.1006/jfan.1993.1107. · Zbl 0799.46053 |
| [8] | W. V. Li and W. Linde, Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab., 27 (1999), pp. 1556-1578, https://doi.org/10.1214/aop/1022677459. · Zbl 0983.60026 |
| [9] | A. A. Borovkov and A. A. Mogul’skii, On probabilities of small deviations for stochastic processes, Siberian Adv. Math., 1 (1991), pp. 39-63. · Zbl 0718.60024 |
| [10] | Q. M. Shao, A small deviation theorem for independent random variables, Theory Probab. Appl., 40 (1996), pp. 191-200, https://doi.org/10.1137/1140021. |
| [11] | M. Ledoux, Isoperimetry and Gaussian analysis, in Lectures on Probability Theory and Statistics (Saint-Flour, 1994), Lecture Notes in Math. 1648, Springer, Berlin, 1996, pp. 165-294, https://doi.org/10.1007/BFb0095676. · Zbl 0874.60005 |
| [12] | W. V. Li and Q.-M. Shao, Gaussian processes: Inequalities, small ball probabilities and applications, in Stochastic Processes: Theory and Methods, Handbook of Statist. 19, North-Holland, Amsterdam, 2001, pp. 533-597, https://doi.org/10.1016/S0169-7161(01)19019-X. · Zbl 0987.60053 |
| [13] | Wenbo V. Li and W. Linde, Existence of small ball constants for fractional Brownian motions, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1329-1334, https://doi.org/10.1016/S0764-4442(98)80189-4. · Zbl 0922.60039 |
| [14] | F. Aurzada and M. A. Lifshits, Small deviations of sums of correlated stationary Gaussian sequences, Theory Probab. Appl., 61 (2017), pp. 540-568, https://doi.org/10.1137/S0040585X97T988356. · Zbl 1377.60048 |
| [15] | A. N. Frolov, A. I. Martikainen, and J. Steinebach, On probabilities of small deviations for compound renewal processes, Theory Probab. Appl., 52 (2008), pp. 328-337, https://doi.org/10.1137/S0040585X97983043. · Zbl 1154.60071 |
| [16] | A. N. Frolov, Limit theorems for small deviation probabilities of some iterated stochastic processes, J. Math. Sci. (N.Y.), 188 (2013), pp. 761-768, https://doi.org/10.1007/s10958-013-1169-0. · Zbl 1282.60039 |
| [17] | O. Zeitouni, Random walks in random environment, in Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1837, Springer, Berlin, 2004, pp. 190-312, https://doi.org/10.1007/978-3-540-39874-5_2. · Zbl 1060.60103 |
| [18] | B. Mallein and P. Miłoś, Brownian motion and random walks above quenched random wall, Ann. Inst. H. Poincaré Probab. Statist., 54 (2018), pp. 1877-1916, https://doi.org/10.1214/17-AIHP859. · Zbl 1417.60072 |
| [19] | B. Mallein, Maximal displacement in a branching random walk through interfaces, Electron. J. Probab., 20 (2015), 68, https://doi.org/10.1214/EJP.v20-2828. · Zbl 1321.60179 |
| [20] | B. Mallein and P. Miłoś, Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment, Stochastic Process. Appl., 129 (2019), pp. 3239-3260, https://doi.org/10.1016/j.spa.2018.09.008. · Zbl 1511.60127 |
| [21] | Y. Lv and W. Hong, On the Barrier Problem of Branching Random Walk in Time-Inhomogeneous Random Environment, preprint, https://arxiv.org/abs/2202.13173, 2022. |
| [22] | Y. Lv, Brownian motion between two random trajectories, Markov Process. Related Fields, 25 (2019), pp. 359-377. · Zbl 1481.60164 |
| [23] | S. Dereich and M. A. Lifshits, Probabilities of randomly centered small balls and quantization in Banach spaces, Ann. Probab., 33 (2005), pp. 1397-1421, https://doi.org/10.1214/009117905000000161. · Zbl 1078.60029 |
| [24] | A. I. Sakhanenko, Estimates in the invariance principle in terms of truncated power moments, Siberian Math. J., 47 (2006), pp. 1113-1127, https://doi.org/10.1007/s11202-006-0119-1. · Zbl 1150.60364 |
| [25] | M. Csörgö and P. Révész, How big are the increments of a Wiener process?, Ann. Probab., 7 (1979), pp. 731-737, https://doi.org/10.1214/aop/1176994994. · Zbl 0412.60038 |
| [26] | A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Appl. Math. (N.Y.) 38, Springer-Verlag, New York, 1998. · Zbl 0896.60013 |
| [27] | K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, 2nd corr. printing, Grundlehren Math. Wiss. 125, Springer-Verlag, Berlin, 1974. · Zbl 0285.60063 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
