The Contribution to the Sum of the Summand of Maximum Modulus

Abstract

Let $X_k$ be i.i.d., $S_n = X_1 + \cdots + X_n$, and $X^{(1)}_n$ the term of maximum modulus among $\{X_1,\ldots, X_n\}$. Let $u_k = P\{2^k < |X_1| \leq 2^{k+1}\parallel |X_1| > 2^k\}$. The main result is that $X^{(1)}_n/S_n \rightarrow 1$ a.s. $\operatorname{iff} \sum u^2_k < \infty$. Furthermore, for any positive integer $r, \lim\inf_{n\rightarrow\infty} |X^{(1)}_n/S_n| = r^{-1} \mathrm{a.s.} \operatorname{iff} \sum_k u^r_k = \infty$ and $\sum_ku^{r+1}_k < \infty$. If $\sum_ku^r_k = \infty$ for all $r$ then $\lim\inf_{n\rightarrow\infty} |X^{(1)}_n/S_n| = 0$ a.s.