Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Precise large deviations for the first passage time of a random walk with negative drift


Authors: Dariusz Buraczewski and Mariusz Maślanka
Journal: Proc. Amer. Math. Soc. 147 (2019), 4045-4054
MSC (2010): Primary 60G50, 60F10
DOI: https://doi.org/10.1090/proc/13632
Published electronically: May 29, 2019
MathSciNet review: 3993796
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S_n$ be partial sums of an i.i.d. sequence $ \{X_i\}$. We assume that $ \mathbb{E} X_1 <0$ and $ \mathbb{P}[X_1>0]>0$. In this paper we study the first passage time

$\displaystyle \tau _u = \inf \{n:\; S_n > u\}.$    

The classical Cramér's estimate of the ruin probability says that

$\displaystyle \mathbb{P}[\tau _u<\infty ] \sim C e^{-\alpha _0 u}$$\displaystyle \qquad \text {as } u\to \infty ,$    

for some parameter $ \alpha _0$. The aim of the paper is to describe precise large deviations of the first crossing by $ S_n$ a linear boundary. More precisely for a fixed parameter $ \rho $ we study asymptotic behavior of $ \mathbb{P}\big [\tau _u = \lfloor u/\rho \rfloor \big ]$ as $ u$ tends to infinity.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60G50, 60F10

Retrieve articles in all journals with MSC (2010): 60G50, 60F10


Additional Information

Dariusz Buraczewski
Affiliation: Instytut Matematyczny, Uniwersytet Wroclawski, 50-384 Wroclaw, pl. Grunwaldzki 2/4, Poland
Email: dbura@math.uni.wroc.pl

Mariusz Maślanka
Affiliation: Instytut Matematyczny, Uniwersytet Wroclawski, 50-384 Wroclaw, pl. Grunwaldzki 2/4, Poland
Email: maslanka@math.uni.wroc.pl

DOI: https://doi.org/10.1090/proc/13632
Keywords: First passage time, ruin problem, large deviations, random walk
Received by editor(s): August 6, 2016
Received by editor(s) in revised form: December 2, 2016
Published electronically: May 29, 2019
Additional Notes: The research was partially supported by the National Science Centre, Poland (Sonata Bis, grant No. UMO-2014/14/E/ST1/00588)
Communicated by: David Levin
Article copyright: © Copyright 2019 American Mathematical Society