A probabilistic representation of constants in Kesten’s renewal theorem

Abstract

Abstract. The aims of this paper are twofold. Firstly, we derive a probabilistic representation for the constant which appears in the one-dimensional case of Kesten's renewal theorem. Secondly, we estimate the tail of a related random variable which plays an essential role in the description of the stable limit law of one-dimensional transient sub-ballistic random walks in random environment.

Limit laws for transient random walks in random environment on \mathbb{Z}

“…-Note that several probabilistic representations are available to compute C K numerically, which are equally efficient. The first one was obtained by Goldie [8], a second was conjectured by Siegmund [19], and we obtained a third one in [6], which plays a central role in the proof of the theorem.…”

“…Therefore, we have, on A ‡ (n) and for all large n, Then, we can use Corollary A.1 and Remark A.1 in [6], that together imply…”

Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions

“…We observe that, if d = 1, and A, B are positive, a formula of this type for C = C + , with equality, is given in [14]. We don't know if such an equality is valid in our setting.…”

Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions

“…L(e x ) H(dz). For any δ > 0, the integrand is bounded by ce −δz for some c > 1 by Potter bounds (20). Combining this with (25) and Lebesgue's Dominated Convergence Theorem we conclude that Observe that there exists β * > 0 such that…”