  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 $\tau=\tau_a=\inf\{n:X_n>a\}$. The main results
presented are two term asymptotic expansions for the joint distribution of
$S_{\tau}$ and $\tau$ and the marginal distribution of $h(S_{\tau}/a,\tau/a)$
in the limit $a\to\infty$. These results are used to study the distribution of
$t$-statistics in sequential experiments with sample size $\tau$, and to remove
bias from confidence intervals based on Anscombe's theorem.
