On the Limit Law of a Random Walk Conditioned to Reach a High Level

TL;DR

This paper studies the asymptotic behavior of a negatively drifting random walk conditioned to reach high levels, showing convergence to a spectrally-positive Lévy process under specific scaling.

Contribution

It introduces a new limit law for conditioned random walks with stable law domain of attraction, extending understanding of their asymptotic properties.

Findings

Convergence of scaled conditioned random walk to a spectrally-positive Lévy process.

Identification of the limit process as a nondecreasing Markov process.

Application of Cramér's change of measure in the analysis.

Abstract

We consider a random walk with a negative drift and with a jump distribution which under Cram\'er's change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally-positive L\'evy %-Khinchin process conditioned not to overshoot level one.