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

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