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Annealed deviations of random walk in random scenery. (English) Zbl 1119.60083
Let \((Z_n)_{n=1,2,\dots}\) be a \(d\)-dimensional random walk in random scenery, i.e., \(Z_n=\sum_{k=0}^{n-1} Y(S_k)\) with \((S_k)_{k=0,1,\dots}\) a random walk in \(\mathbb{Z}^d\) and \((Y(z))_{z \in \mathbb{Z}^d}\) an i.i.d. scenery, independent of the walk. The walker’s step has mean zero and some finite exponential moments. The speed and the rate of \(\log P(n^{-1}Z_n >b_n)\) are identified for various choices of sequences \((b_n)_{n=1,2,\dots}\). For example, the so-called very large deviations are defined by requirements \(1 \ll b_n\ll n^{1/q}\) for some \(q>0\). Depending on \((b_n)_n\) and the upper tails of the scenery, the authors identify different regimes for the speed of \(\log P(n^{-1}Z_n >b_n)\) and different variational formulas for the rate functions. The considered problem has been previously addressed in the work of A. Asselah and F. Castell [Probab. Theory Relat. Fields 126, No. 4, 497–527 (2003; Zbl 1043.60018)]. The main novelty of the paper is the study of arbitrary sceneries unbounded to \(+\infty\) and general scale functions \(b_n \gg 1\) in the discrete setting.
The results of the article have interesting connections to large deviation properties of self-intersections of the walk, which have been studied by X. Chen [Ann. Probab. 32, No. 4, 3248–3300 (2004; Zbl 1067.60071)].

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