Some Blackwell-Type Renewal Theorems for Weighted Renewal Functions

Lin, Jianxi

doi:10.1017/s0021900200004927

Cited by 1 publication

(4 citation statements)

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“…Of the known general results, the following assertion stated as Theorem 6.1 in [15] is the closest one to the main topic of the present paper: If the sequence {a n } is regularly oscillating in the sense that lim…”

Section: Introductionmentioning

confidence: 76%

“…A relation analogous to (1.4) holds under the same assumptions in the arithmetic case as well (Theorem 3.1 in [15]). In fact, it were these results that drew our attention to the problem on the asymptotic behaviour of h(x, ∆) as x → ∞, as we realized that the conditions imposed in [15] on the weight sequence {a n } could be substantially relaxed provided that F belongs to the domain of attraction of a stable law.…”

Section: Introductionmentioning

confidence: 78%

“…To bound Σ 2± we will need the following extension of Lemma 6.1 from [15] to the case where P(ξ < 0) > 0. Set…”

Section: Now Letmentioning

confidence: 99%

“…There is substantial literature devoted to studying the asymptotics of H(x) and h(x, ∆) as x → ∞, recent surveys of the area and some further references being available in [20,15]. In what follows, we will mostly be mentioning results referring to h(x, ∆).…”

Section: Introductionmentioning

confidence: 99%

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Blackwell-type theorems for weighted renewal functions

Боровков

Borovkov

2014

Sib Math J

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For a numerical sequence {a n } satisfying broad assumptions on its "behaviour on average" and a random walk S n = ξ 1 +· · ·+ξ n with i.i.d. jumps ξ j with positive mean µ, we establish the asymptotic behaviour of the sums n 1 a n P(S n ∈ [x, x + ∆)) as x → ∞, where ∆ > 0 is fixed. The novelty of our results is not only in much broader conditions on the weights {a n }, but also in that neither the jumps ξ j nor the weights a j need to be positive. The key tools in the proofs are integro-local limit theorems and large deviation bounds. For the jump distribution F , we consider conditions of four types: (a) the second moment of ξ j is finite, (b) F belongs to the domain of attraction of a stable law, (c) the tails of F belong to the class of the so-called locally regularly varying functions, (d) F satisfies the moment Cramér condition. Regarding the weights, in cases (a)-(c) we assume that {a n } is a so-called ψ-locally constant on average sequence, ψ(n) being the scaling factor ensuring convergence of the distributions of (S n − µn)/ψ(n) to the respective stable law. In case (d) we consider sequences of weights of the form a n = b n e qn , where {b n } has the properties assumed about the sequence {a n } in cases (a)-(c) for ψ(n) = √ n.

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 78%

“…To bound Σ 2± we will need the following extension of Lemma 6.1 from [15] to the case where P(ξ < 0) > 0. Set…”

Section: Now Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations