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Random walks crossing curved boundaries: a functional limit theorem, stability and asymptotic distributions for exit times and positions

Part of: Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

R. A. Doney  and
R. A. Maller
R. A. Doney*
Affiliation:
University of Manchester
R. A. Maller*
Affiliation:
University of Western Australia
*
∗ Postal address: Department of Mathematics, University of Manchester, Manchester M13 9PL, UK. Email address: rad@fs2.ma.man.ac.uk
∗∗ Postal address: Department of Accounting and Finance, The University of Western Australia, Nedlands 6907, Australia.
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Abstract

We study the (two-sided) exit time and position of a random walk outside boundaries which are regularly varying functions of smaller order at infinity than the square root. A natural domain of interest is those random walks which are attracted without centring to a normal law, or are relatively stable. These are shown to have ‘stable’ exit positions, in that the overshoot of the curved boundary is of smaller order of magnitude (in probability) than the boundary, as the boundary expands. Surprisingly, this remains true regardless of the shape of the boundary. Furthermore, within the same natural domain of interest, norming of the exit position by, for example, the square root of the exit time (in the finite-variance case), produces limiting distributions which are computable from corresponding functionals of Brownian motion. We give a functional limit theorem for attraction of normed sums to general infinitely divisible random variables, as a means of making this, and more general, computations. These kinds of theorems have applications in sequential analysis, for example.

Keywords

Random walkexit timeexit positioncurved boundaryfunctional limit theoremdomain of attraction of the normalrelative stability

MSC classification

Primary: 60K05: Renewal theory 60F05: Central limit and other weak theorems
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1117 - 1149
Copyright
Copyright © Applied Probability Trust 2000 

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