A note on the variance of a stopping time (Q1098489)

Let \(\{S_ n=\sum^{n}_{1}X_ i\}_{n\geq 0}\) be a random walk with \(\mu =E X_ i>0\) and \(\sigma\) \(2=Var X_ i<\infty\). Let \(\tau (b)=\inf \{n\geq 1:\) \(S_ n>b\}\) be the hitting time of the set (b,\(\infty)\). \textit{T. L. Lai} and \textit{D. Siegmund} [Ann. Stat. 7, 60-76 (1979; Zbl 0409.62074)] showed that Var \(\tau\) (b)\(=b \sigma\) 2/\(\mu\) \(3+K/\mu\) \(2+o(1)\) as \(b\to \infty\), and gave an expression for the constant K which is difficult to compute. In this paper, the expression for K is simplified to a form that depends only on moments of ladder variables.