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Deviations of a random walk in a random scenery with stretched exponential tails. (English) Zbl 1100.60056

Consider a \(d\)-dimensional random walk in a random scenery \(Z_n =\sum_{k=0}^{n-1} Y_{S_k}\), \(n\geq 1\), where \((S_k; k\in \mathbb N)\) is a random walk in \(\mathbb Z^d\) starting at the origin and \((Y_x;x\in\mathbb Z^d)\) is an i.i.d. scenery, independent of the walk. The assumptions on the scenery are that \(Y_0\) is centered with a finite second moment and has a stretched exponential tail. In particular, \(Y_0\) does not have any positive exponential moments. Under these assumptions, it is known that for \(d\geq 1\), there is a sequence \((a_n)\) such that \(Z_n/na_n\) converges in distribution towards some non-degenerate random variable. For \(d\geq3\), \(a_n =n^{-1/2}\). The goal of this paper is to obtain precise logarithmic asymptotics for \( \mathbb{P}(Z_n > n t_n)\) for some sequences of positive numbers \((t_n)\) such that \(\lim_n t_n/a_n = \infty\). The first step of the proof is a generalization of a result of A. V. Nagaev [Teor. Veroyatn. Primen. 14, 203-216 (1969; Zbl 0181.45004) and ibid. 14, 51-63 (1969; Zbl 0172.21901)]. Then, using moderate deviations for the local time of the random walk at the origin, the proof can be completed.

MSC:

60K37 Processes in random environments
60F10 Large deviations
60J55 Local time and additive functionals
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References:

[1] Asselah, A.; Castell, F., Large deviations for Brownian motion in a random scenery, Probab. Theory Relat. Fields, 126, 497-527 (2003) · Zbl 1043.60018
[2] Asselah, A.; Castell, F., Quenched large deviations for diffusions in a random Gaussian shear flow drift, Stochastic Proc. Appl., 103, 1, 1-29 (2003) · Zbl 1075.60508
[3] A. Asselah, F. Castell, A note on random walk in random scenery, preprint, 2005.; A. Asselah, F. Castell, A note on random walk in random scenery, preprint, 2005. · Zbl 1112.60088
[4] Bolthausen, E., A central limit theorem for two-dimensional random walks in random sceneries, Ann. Probab., 17, 108-115 (1989) · Zbl 0679.60028
[5] Castell, F., Moderate deviations for diffusions in a random Gaussian shear flow drift, Ann. Inst. H. Poincaré Probab. Statist., 40, 3, 337-366 (2004) · Zbl 1042.60009
[6] Castell, F.; Pradeilles, F., Annealed large deviations for diffusions in a random shear flow drift, Stochastic Proc. Appl., 94, 171-197 (2001) · Zbl 1051.60028
[7] Chen, X., Moderate deviations for Markovian occupation times, Stochastic Proc. Appl., 94, 51-70 (2001) · Zbl 1051.60029
[8] Dembo, A.; Zeitouni, O., Large Deviations Techniques and Applications (1998), Springer: Springer Berlin · Zbl 0896.60013
[9] Eichelsbacher, P.; Löwe, M., Moderate deviations for i.i.d. random variables, ESAIM: Probab. Statist., 7, 209-218 (2003) · Zbl 1019.60021
[10] N. Gantert, W. König, Z. Shi, Annealed deviations of random walk in random scenery, preprint, 2005.; N. Gantert, W. König, Z. Shi, Annealed deviations of random walk in random scenery, preprint, 2005.
[11] Gantert, N.; Zeitouni, O., Large and moderate deviations for the local time of a recurrent Markov chain, Ann. Inst. H. Poincaré Probab. Statist., 34, 5, 687-704 (1998) · Zbl 0910.60013
[12] Gärtner, J.; König, W., The parabolic Anderson model, (Deuschel, J.-D.; Greven, A., Interacting Stochastic Systems (2005), Springer: Springer Berlin), 153-179 · Zbl 1111.82011
[13] Kesten, H.; Spitzer, F., A limit theorem related to a new class of self-similar processes, Z. Wahrsch. Verw. Geb., 50, 5-25 (1979) · Zbl 0396.60037
[14] Nagaev, A. V., Integral limit theorems for large deviations when Cramer’s condition is not fulfilled, I, II, Theory Probab. Appl., 14, 51-64 (1969), 193-208 · Zbl 0196.21002
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