Small drift limit theorems for random walks

Conditioned Limit Theorems for Random Walks with Negative Drift*

In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant c~ so that (St,.j/c~nl/2[N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES~ = a < 0 , ES~ < oo, and there is a slowly var...

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 ...

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...

Ratio Limit Theorems for Random Walks on Groupso