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Abstract
We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha$. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long standing problem, which dates back to the 1962 paper of Garsia and Lamperti [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790
Highlights
Introduction and resultsThis paper contains new results about asymptotically stable random walks
We denote by RV (γ) the class of regularly varying functions with index γ, namely f ∈ RV (γ) if and only if f (x) = xγl(x) for some slowly varying function l ∈ RV (0), see [BGT89]
We assume (4.1), which is equivalent to the (SRT), and we deduce that I1(δ; x) is a.n. and, for any k ≥ 2, that Ik(δ, η; x) is a.n., for every fixed η ∈ (0, 1)
Summary
We first present a local large deviation estimate which improves the error term in the classical local limit theorems, without making any further assumptions (see Theorem 1.1) We exploit this bound to solve a long-standing problem, namely we establish necessary and sufficient conditions for the validity of the strong renewal theorem (SRT), both for renewal processes (Theorem 1.4) and for general random walks (Theorem 1.12). For a more usual formulation, we can write A(x) = xα/L(x) with L(·) slowly varying: This relation, called strong renewal theorem (SRT), is known to follow from (1.2) when α. In this paper we settle this problem, determining necessary and sufficient conditions for the SRT : see Theorem 1.4 for renewal processes and Theorem 1.12 for random walks.
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