Tail sums of convergent series of independent random variables

Abstract

Let X 1 , X 2 , … be a sequence of independent random variables such that, for each n ≥ 1, EX n = 0 and and assume that then converges almost surely as N → ∞. Let and let F n ( x ) denote the distribution function of X n . Loynes (2) observed that the sequence { S n } is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence { S n }, and hence the way in which converges to its limit.

Locations

• Mathematical Proceedings of the Cambridge Philosophical Society - View
• Zurich Open Repository and Archive (University of Zurich) - View - PDF