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.