Average extrema of a random walk with a negative binomial stopping time (Q2867737)

The author considers a simple random walk that moves upward and downward one unit at a time with probability \(p\) and \(1-p\), respectively. Let \(T_{n}\) denote the number of steps taken upon making \(n\) downward movements. Let \(M_{i}\) and \(m_{i}\) be the maximum and minimum heights obtained through \(i\) steps. The paper deals mainly with the expectations \(\operatorname{E}M_{T_{n}}\), \(\operatorname{E}m_{T_{n}-1}\) and \(\operatorname{E}m_{T_{n}}\). A version of a random walk that moves up and down \(j\) units at a time is also considered.