Limit Theorems for Randomly Selected Partial Sums

Abstract

Let $\{S_n\}$ be the partial sums of a sequence (not necessarily independent) of random variables $\{X_n\}$, and let $\{m_u\}$ be a set of integer-valued random variables depending on an index $u \geqq 0$. Suppose that $m_u/u$ converges in probability to a constant as $u \rightarrow \infty$ and that $S_n$ 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 $S_n$ do not fluctuate too much, in a sense made precise below, then the random sum $S_{m_u}$ 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 $\{X_n\}$ satisfies the invariance principle then $S_{m_u}$ is asymptotically normal. The invariance principle has been proved in [2] for various dependent sequences $\{X_n\}$, 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 $\max_{k \leqq m_u} S_k$. In Section 4, these theorems are applied to renewal processes.