Local asymptotics for the first intersection of two independent renewals

Abstract

We study the intersection of two independent renewal processes, ρ = τ ∩σ. Assuming that P(τ 1 = n) = ϕ(n) n −(1+α) and P(σ 1 = n) = ϕ(n) n −(1+ α) for some α, α 0 and some slowly varying ϕ, ϕ, we give the asymptotic behavior first of P(ρ 1 > n) (which is straightforward except in the case of min(α, α) = 1) and then of P(ρ 1 = n). The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities P(ρ 1 = n) while knowing asymptotically the renewal mass function P(n ∈ ρ) = P(n ∈ τ)P(n ∈ σ). Our results can be used to bound coupling-related quantities, specifically the increments |P(n ∈ τ) − P(n − 1 ∈ τ)| of the renewal mass function. 1. Intersection of two independent renewals We consider two independent (discrete) renewal processes τ and σ, whose law are denoted respectively P τ and P σ , and the renewal process of intersections, ρ = τ ∩ σ. We denote P = P τ ⊗ P σ. The process ρ appears in various contexts. In pinning models, for example, it may appear directly in the definition of the model (as in [1], where σ represents sites with nonzero disorder values, and τ corresponds to the polymer being pinned) or it appears in the computation of the variance of the partition function via a replica method (see for example [20]), and is central in deciding whether disorder is relevant or irrelevant in these models, cf. [3]. When τ and σ have the same inter-arrival distribution, ρ 1 is related to the coupling time of τ and σ, if we allow τ and σ to start at different points. In particular, in the case µ := E[τ 1 ] < +∞, the coupling time ρ 1 has been used to study the rate of convergence in the renewal theorem, see [16, 17], using that