TY - JOUR

T1 - Random walks crossing curved boundaries

T2 - a functional limit theorem, stability and asymptotic distributions for exit times and positions

AU - Doney, R. A.

AU - Maller, R. A.

PY - 2000/12

Y1 - 2000/12

N2 - 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 centering 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.

AB - 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 centering 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.

UR - http://www.scopus.com/inward/record.url?scp=0034428648&partnerID=8YFLogxK

U2 - 10.1017/S0001867800010491

DO - 10.1017/S0001867800010491

M3 - Article

SN - 0001-8678

VL - 32

SP - 1117

EP - 1149

JO - Advances in Applied Probability

JF - Advances in Applied Probability

IS - 4

ER -