Asymptotics for the First Passage Times of Lévy Processes and Random Walks

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically α-stable Lévy processes with α < 1.Our main result states that if the left tail of the Lévy measure is regularly varying with index −α and the moving boundary is equal to 1 − t γ for some γ < 1/α, then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary 1 + t γ with γ < 1/α under the assumption of a regularly varying right tail with index −α.

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“…right) tail is used. Hence, our proof gives some hope to be generalized to other Lévy processes such as processes indicated in [15].…”

Section: Introduction and Main Resultsmentioning

confidence: 86%

“…[22]). If the process does not necessarily satisfy Spitzer's condition, various results were obtained for a constant boundary by [4,6,10,11,15,18,28].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

Аурзада

Kramm

2015

J Theor Probab

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“…That is, how does the random walk behave asymptotically, conditioned it stays above a boundary sequence that is not necessarily zero or even constant? This topic, as well as the closely related first-passage asymptotics, has been addressed in [15,16,11,12,5,1,4] and many more.…”

Section: Introductionmentioning

confidence: 99%

First-passage time asymptotics over moving boundaries for random walk bridges

Sloothaak¹,

Wachtel

Zwart

2018

J. Appl. Probab.

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We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent −1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics, of which the precise nature depends on the order of distance between zero and the moving boundary.

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On busy periods of the critical GI/G/1 queue and BRAVO

Nazarathy

Palmowski

2022

Queueing Syst

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We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of $$2+\epsilon $$ 2 + ϵ moments.

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Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Cited by 8 publications

References 29 publications

The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

First-passage time asymptotics over moving boundaries for random walk bridges

On busy periods of the critical GI/G/1 queue and BRAVO

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