Weak convergence of conditioned sums of independent random vectors.

Abstract:

Conditions are given for the weak convergence of processes of the form $({{\mathbf {X}}_n}(t)|{{\mathbf {X}}_n}(1) \in {E^n})$ to tied-down stable processes, where $({{\mathbf {X}}_n}(t)$ is constructed from normalized partial sums of independent and identically distributed random vectors which are in the domain of attraction of a multidimensional stable law. The conditioning events are defined in terms of subsets ${E^n}$ of ${R^d}$ which converge in an appropriate sense to a set of measure zero. Assumptions which the sets ${E^n}$ must satisfy include that they can be expressed as disjoint unions of “asymptotically convex” sets. The assumptions are seen to hold automatically in the special case in which ${E^n}$ is taken to be a “natural” neighborhood of a smooth compact hypersurface in ${R^d}$.