Tail estimates for one-dimensional random walk in random environment. (English) Zbl 0868.60058
Summary: Suppose that the integers are assigned i.i.d. random variables \(\{\omega_x\}\) (taking values in the unit interval), which serve as an environment. This environment defines a random walk \(\{X_k\}\) (called an RWRE) which, when at \(x\), moves one step to the right with probability \(\omega_k\), and one step to the left with probability \(1-\omega_x\). F. Solomon [Ann. Probab. 3, 1-31 (1975; Zbl 0305.60029)] determined the almost-sure asymptotic speed (= rate of escape) of an RWRE. For certain environment distributions where the drifts \(2\omega_x-1\) can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like \(\exp(-Cn^{1/3})\). This differs sharply from the rates derived by A. Greven and F. den Hollander [ibid. 22, No. 3, 1381-1428 (1994; Zbl 0820.60054)] for large deviation probabilities conditioned on the environment. As a by-product we also provide precise tail and moment estimates for the total population size in a branching process with random environment.
Keywords:
random walk; almost-sure asymptotic speed; rate of escape; large-deviation probabilities decay; branching process with random environmentReferences:
| [1] | Athreya, K.B., Karlin, S.: ”On branching processes with random environments: I. Extinction probabilities”. Ann. Math. Stat.42, 1499–1520 (1971) · Zbl 0228.60032 · doi:10.1214/aoms/1177693150 |
| [2] | Chung, K.L.: Markov Chains with Stationary Transition Probabilities. Berlin: Springer, 1960 · Zbl 0092.34304 |
| [3] | Dembo, A., Karlin, S.: ”Strong limit theorems of empirical functionals for large exceedances of partial sums of i.i.d. variables”. Ann. Probab.19, 1737–1755 (1991) · Zbl 0746.60028 · doi:10.1214/aop/1176990232 |
| [4] | Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Boston: Jones and Bartlett, 1993 · Zbl 0793.60030 |
| [5] | Greven, A., den Hollander, F.: ”Large deviations for a random walk in random environment”. Ann. Probab.22, 1381–1428 (1994) · Zbl 0820.60054 · doi:10.1214/aop/1176988607 |
| [6] | Gut, A.: Stopped random walks. Berlin: Springer, 1988 · Zbl 0634.60061 |
| [7] | Kesten, H.: ”The limit distribution of Sinai’s random walk in random environment”. Phys. A.138, 299–309 (1986) · Zbl 0666.60065 · doi:10.1016/0378-4371(86)90186-X |
| [8] | Kesten, H., Kozlov, M.V., Spitzer, F.: ”A limit law for random walk in a random environment”. Comp. Math.30, 145–168 (1975) · Zbl 0388.60069 |
| [9] | Nagaev, S.V.: ”Large deviations of sums of independent random variables”. Ann. Probab.7, 745–789 (1979) · Zbl 0418.60033 · doi:10.1214/aop/1176994938 |
| [10] | Sinai, Ya.G.: ”The limiting behavior of a one dimensional random walk in a random medium”. Theory Probab. Appl.27, 256–268 (1982) · Zbl 0505.60086 · doi:10.1137/1127028 |
| [11] | Solomon, F.: ”Random walks in random environment”. Ann. Probab.3, 1–31 (1975) · Zbl 0305.60029 · doi:10.1214/aop/1176996444 |
| [12] | Spitzer, F.: Principles of random walk. Berlin: Springer, 1976 · Zbl 0359.60003 |
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.
