The Annals of Probability

Boundary Crossing Problems for Sample Means

Tze Leung Lai

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Motivated by several classical sequential decision problems, we study herein the following type of boundary crossing problems for certain nonlinear functions of sample means. Let $X_1, X_2,\ldots$ be i.i.d. random vectors whose common density belongs to the $k$-dimensional exponential family $h_\theta(x) = \exp\{\theta'x - \psi(\theta)\}$ with respect to some nondegenerate measure $\nu$. Let $\bar{X}_n = (X_1 + \cdots + X_n)/n, \hat\theta_n = (\nabla\psi)^{-1}(\bar{X}_n)$, and let $I(\theta, \lambda) = E_\theta\log\{h_\theta(X_1)/h_\lambda(X_1)\}$ ( = Kullback-Leibler information number). Consider stopping times of the form $T_c(\lambda) = \inf\{n: I(\hat\theta_n, \lambda) \geq n^{-1}g(cn)\}, c > 0$, where $g$ is a positive function such that $g(t) \sim \alpha \log t^{-1}$ as $t \rightarrow 0$. We obtain asymptotic approximations to the moments $E_\theta T^r_c(\lambda)$ as $c \rightarrow 0$ that are uniform in $\theta$ and $\lambda$ with $|\lambda - \theta|^2/c \rightarrow \infty$. We also study the probability that $\bar{X}_{Tc(\lambda)}$ lies in certain cones with vertex $\nabla\psi (\lambda)$. In particular, in the one-dimensional case with $\lambda > \theta$, we consider boundary crossing probabilities of the form $P_\theta\{\hat\theta_n \geq \lambda \text{and} I(\hat\theta_n, \lambda) \geq n^{-1} g(cn) \text{for some} n\}$. Asymptotic approximations (as $c \rightarrow 0$) to these boundary crossing probabilities are obtained that are uniform in $\theta$ and $\lambda$ with $|\lambda - \theta|^2/c \rightarrow \infty$.

Article information

Ann. Probab., Volume 16, Number 1 (1988), 375-396.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F10: Large deviations
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62L05: Sequential design 62L15: Optimal stopping [See also 60G40, 91A60]

Sequential analysis stopping time boundary crossing probabilities exponential family Kullback-Leibler information uniform strong laws


Lai, Tze Leung. Boundary Crossing Problems for Sample Means. Ann. Probab. 16 (1988), no. 1, 375--396. doi:10.1214/aop/1176991909.

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