On the Asymptotic Normality of One-sided Stopping Rules

Abstract

Assume that $x_1, x_2, \cdots$ are independent random variables with expectation $\mu > 0$ and finite variance $\sigma^2$. Let $s_k = x_1 + \cdots + x_k (k = 1, 2, \cdots)$. For any family of positive, non-decreasing, eventually concave functions $f_c$ defined on the positive real numbers and indexed by $c > 0$ such that $f_c \rightarrow \infty$ as $c \rightarrow \infty$, define \begin{equation*}\begin{align*}\tau &= \tau(c) = \text{first}\quad k \geqq 1\quad\text{such that}\quad s_k > f_c(k) \\ &= \infty\quad\text{if no such}\quad k\quad\text{exists} \end{align*}.\end{equation*} The stopping variable (sv) $\tau$ arises in various problems in probability and statistics. For example, the sequential statistical procedures of [2], [3], and [8] involve sv's similar to $\tau$ (see also the next to last paragraph of this section). Suppose that the family $\{f_c:c > 0\}$ is such that we may define $\lambda = \lambda(c)$ by \begin{equation*}\tag{1}\mu\lambda = f_c(\lambda). \end{equation*} In [9] it is shown under conditions on the joint distribution of $x_1, x_2, \cdots$ weaker than the above that \begin{equation*}\tag{2}E\tau \sim\lambda\quad(c \rightarrow \infty)\end{equation*} for a certain class of families $\{f_c\}$. In Section 2 of this note it is shown that if $f_c(x) = cx^\alpha$ for some $0 \leqq \alpha < 1$, then $\tau$ (suitably normalized) is asymptotically normally distributed whenever $(s_n - n\mu)/\sigma n^{\frac{1}{2}}$ is. (See also remarks (a) and (b) in Section 4.) This extends Heyde's result [5], [7], valid when the $x_k$ have a common distribution and $\alpha = 0$. Assume now for simplicity that $f_c(x) \equiv c$ and $x_1, x_2, \cdots$ have a common distribution. If $x_1 \geqq 0$ the random variable $M(c)(N(c))$ defined by \begin{equation*}\tag{3}M(c) = \sup\{n:s_n \leqq c\} (N(c) = \sum^\infty_{n = 1}I\{s_n \leqq c\})\end{equation*} is of interest in renewal theory; and the observation that $M(c) = N(c) = \tau(c) - 1$ allows one to study $M(N)$ by studying the stopping variable $\tau$. In the general case it has been noted by several authors that the behavior of $\tau$ and $N$ may differ in important respects. (For example, if $E(x_1^-)^2 = \infty$, it is known that $EN = \infty$ [6], whereas it is easy to show that $E\tau < \infty, E\tau \sim c/\mu$ (e.g. [9]).) In Section 3 we point out that some knowledge of $M(N)$ can be obtained in a direct fashion from relevant knowledge of $\tau$. Our methods throughout involve finding convenient estimates for the probability that $s_k$ crosses the curve $f_c$ for the first time at some index $k < n$ and then falls back below the curve at time $n$.