Vague convergence of sums of independent random variables

Abstract

A sequence (μn) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμn →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ0 denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e−1] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ e [0,λ]. In the independent identically distributed case, it is shown that if (x1 + ... +xn)/an, an → ∞, converges vaguely toμ with 0 x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ e (0, 1) one can choosean → ∞ so that the limit measure=βδ0.