Some One-Sided Stopping Rules

Abstract

Suppose that $x_1, x_2, \cdots$ are random variables with means $\mu_1, \mu_2, \cdots$ such that \begin{equation*}\tag{1}n^{-1}(\mu_1 + \cdots + \mu_n) \rightarrow \mu\quad (0 < \mu < \infty),\end{equation*} and let $s_n = x_1 + \cdots + x_n (n \geqq 1)$. For any positive non-decreasing and eventually concave function $f$ defined on the positive real numbers and $c > 0$ let $\tau = \tau(c) = \text{first}\quad n \geqq 1\quad\text{such that}\quad s_n > cf(n)\\= \infty\quad\text{if no such}\quad n\quad\text{exists}.$ We are interested in finding conditions on $f$ and on the joint distribution of $(x_n)$ which insure that if $\lambda = \lambda(c)$ is defined by \begin{equation*}\tag{2}\mu\lambda = cf(\lambda)\end{equation*} (since we shall assume below that $f(n) = o(n), \lambda(c)$ is unique for sufficiently large $c$), then \begin{equation*}\tag{3}\lim_{c \rightarrow \infty}\lambda^{-1}E\tau = 1.\end{equation*} The elementary renewal theorem states that (3) holds when $x_1, x_2, \cdots$ are iid non-negative random variables and $f(n) \equiv 1$. Chow and Robbins [3] have obtained generalizations of this result to the case in which $(x_n)$ are dependent or non-identically distributed. The case in which $f$ is not constant has been discussed in [2], [4], and [11]. We shall assume that $x_1, x_2, \cdots$ are independent and prove the Theorem. Let $(x_n), (\mu_n), f, \tau, c, \lambda$ be as above, and suppose that for some $\alpha\varepsilon(0, 1)$ and $L$ slowly varying \begin{equation*}\tag{4}f(n) \sim n^\alpha L(n).\end{equation*} If for every $\epsilon > 0$ \begin{equation*}\tag{5}n^{-1} \sum^n_1 \int_{\{x_i - \mu_i > i\epsilon\}} (x_i - \mu_i) \rightarrow 0,\quad n \rightarrow \infty,\end{equation*} then \begin{equation*}\tag{6}\lim\sup\lambda^{-1} E \tau \leqq 1.\end{equation*} If in addition to (5) $s_n/n \rightarrow \mu$ a.s. or if for every $\epsilon > 0$ \begin{equation*}\tag{7}n^{-1} \sum^n_1 \int_{\{|x_i - \mu_i| > n\epsilon\}} |x_i - \mu_i| \rightarrow 0,\quad n \rightarrow \infty,\end{equation*} \begin{equation*}\tag{8}\sup_n E |x_n - \mu_n| \leqq K < \infty,\end{equation*} then (3) holds.