Asymptotic expansions in anscombe's theorem for repeated significance tests and estimation after sequential testing

Let x1,x2,⋯ be independent and normally distributed with unknown mean θ and variance 1. Let τ=inf{n≥1:|sn|≥√2a(n+c)}. Then a repeated significance test for a normal mean rejects the hypothesis θ=0 if and only if τ≤N0 for some positive integer N0. The problem we consider is estimation of θ based on the data x1,⋯,xT,T=min{τ,N0}. We shall solve this problem by obtaining the asymptotic expansion of the distribution of (sτ−τθ)/√τ as a→∞, and then constructing the confidence intervals for θ.