# Vague convergence of sums of independent random variables

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## Abstract

A sequence (*μ* _{n}) of probability measures on the real line is said to converge vaguely to a measure*μ* if*∫ fdμ* _{n} →*∫ fdμ* for every continuous function*f* with*compact* support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖*μ*‖ denote the total mass of*μ* and*δ* _{0} 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 measure*μ* there is a closed interval [0,*λ*], [0,*e* ^{−1}] ⊂ [0,*λ*] ⊂ [0, 1], such that*βμ* is attainable as a limit of infinitely divisible probability laws iff*β* ε [0,*λ*]. In the independent identically distributed case, it is shown that if (*x* _{1} + ... +*x* _{n})/*a* _{n}, a_{n} → ∞, converges vaguely to*μ* with 0<‖*μ*‖<1, then*μ*=‖*μ*‖*δ* _{0}. If furthermore the ratios*a* _{ n+1}/*a* _{n} are bounded above and below by positive numbers, then*L(x)*=*P*[|*X* _{1}|>*x*] is a slowly varying function of*x*. Conversely, if*L(x)* is slowly varying, then for every*β* ε (0, 1) one can choose*a* _{n} → ∞ so that the limit measure=*βδ* _{0}.

## Keywords

Probability Measure Independent Random Variable Complete Convergence Triangular Array Centered Array## Preview

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## References

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