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Random walks crossing curved boundaries: A functional limit theorem, stability and asymptotic distributions for exit times and positions. (English) Zbl 0976.60082

Authors’ 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.
Reviewer: A.Gut (Uppsala)

MSC:

60K05 Renewal theory
60F05 Central limit and other weak theorems
60G40 Stopping times; optimal stopping problems; gambling theory
60G50 Sums of independent random variables; random walks
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