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Transactions of the American Mathematical Society

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Weak convergence of conditioned sums of independent random vectors.

Author: Thomas M. Liggett
Journal: Trans. Amer. Math. Soc. 152 (1970), 195-213
MSC: Primary 60.30
MathSciNet review: 0268940
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Abstract: Conditions are given for the weak convergence of processes of the form $ ({{\mathbf{X}}_n}(t)\vert{{\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}$.

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Keywords: Weak convergence, random vectors, stable processes, conditioned processes
Article copyright: © Copyright 1970 American Mathematical Society

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