Local limit theorems and renewal theory with no moments

Abstract : We study renewals τ with index 0: the inter-arrival distribution is P(τ 1 = n) = ϕ(n)n −1 , with ϕ(·) slowly varying. We obtain a strong renewal theorem, that is P(n ∈ τ) n→∞ ∼ P(τ 1 = n)/P(τ 1 ≥ n) 2. If instead we only assume regular variation of P(n ∈ τ) and slow variation of U n := n k=0 P(k ∈ τ), we obtain a similar equivalence but with P(τ 1 = n) replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. Along the way we prove a local limit theorem and a local (upward) large deviation theorem, giving the asymptotics of P(τ k = n) when n is at least the typical length of τ k. We further derive downward moderate and large deviations estimates, that is, the asymptotics of P(τ k ≤ n) when n is much smaller than the typical length of τ k .
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Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21 (66), <10.1214/16-EJP13>
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Kenneth S. Alexander, Quentin Berger. Local limit theorems and renewal theory with no moments. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21 (66), <10.1214/16-EJP13>. <hal-01291344v2>

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