arXiv subjects
Publications and source records attributed to Robert Keener.
Donsker's theorem shows that random walks behave like Brownian motion in an asymptotic sense. This result can be used to approximate expectations associated with the time and location of a random walk when it first crosses a nonlinear boundary. In this paper, correction terms are derived to improve the accuracy of these approximations.
Let $\{S_n=(X_n,W_n)\}_{n\ge0}$ be a random walk with $X_n\in \mathbb{R}$ and $W_n\in \mathbb{R}^m$. Let $τ=τ_a=\inf\{n:X_n>a\}$. The main results presented are two term asymptotic expansions for the joint distribution of $S_τ$ and $τ$ and the marginal distribution of $h(S_τ/a,τ/a)$ in the limit $a\to\infty$. These results are used to study the distribution of $t$-statistics in sequential experiments with sample size $τ$, and to remove bias from confidence intervals based on Anscombe's theorem.