Limit Theorems for Randomly Selected Partial Sums

Let ₙ\ be the partial sums of a sequence (not necessarily independent) of random variables ₙ\, and let ᵤ\ be a set of integer-valued random variables depending on an index u 0. Suppose that mᵤ/u converges in probability to a constant as u → ∞ and that Sₙ obeys the central limit theorem (when it is normed properly, as must also be the other variables below). Anscombe [1] has shown that if the Sₙ do not fluctuate too much, in a sense made precise below, then the random sum Smᵤ also obeys the central limit theorem. Anscombe's condition is closely related to one introduced by Prohorov [6] in connection with the Erdos-Kac-Donsker invariance principle. In Section 2 this relationship is investigated; in particular, it is shown that if the sequence ₙ\ satisfies the invariance principle then Smᵤ is asymptotically normal. The invariance principle has been proved in [2] for various dependent sequences ₙ\, to each of which this result is then applicable. In Section 3 an invariance principle is formulated and proved for the random partial sums; this result enables one to find, for example, the limiting distribution of maxk mᵤ Sₖ. In Section 4, these theorems are applied to renewal processes.