Limit theorems for additive functionals of random walks in random scenery

Quenched Central Limit Theorems for Random Walks in Random Scenery

When the support of X1 is a subset of N , (Sn)n≥0 is called a renewal process. Each time the random walk is said to evolve in Z, it implies that the walk is truly d-dimensional, i.e. the linear space generated by the elements in the support of X1 is d-dimensional. Institut Camille Jordan, CNRS UMR 5208, Université de Lyon, Université Lyon 1, 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne...

Renewal theorems for random walks in random scenery∗

Random walks in random scenery are processes defined by Zn := ∑n k=1 ξX1+...+Xk , where (Xk, k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We suppose that the distributions of X1 and ξ0 belong to the normal domain of attraction of strictly stable distributions with index α ∈ [1, 2] and β ∈ (0, 2] respectively. We are interested in the asymptotic behaviour as |...

Limit theorems for one and two-dimensional random walks in random scenery

Limit Theorems for Random Walks with Boundaries

In this review, we consider boundary problems for random walks generated by sums of independent items and some of their generalizations. Let 1, 42, . . * be identically distributed independent random variables with distribution frunction F(x). Let S = 0, Sn = Sk= Ok with n = 1, 2, * -. We shall study the properties of the random trajectory formed by the sums S0, S1, 82, . Let n be an integer pa...

Conditional limit theorems for ordered random walks

In a recent paper of Eichelsbacher and König (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a k-dimensional random walk conditioned to stay in a strict order at all times. Moreover, they have shown that the rescaled random walk converges to the Dyson Brownian motion. In the present paper we find the ...