Some exact distributions of the number of one-sided deviations and the time of the last such deviation in the simple random walk (Q1091669)

Let \(S_ n\) be the n-th partial sum of i.i.d. Bernoulli r.v. with probabilities p \((0<p<1)\) and \(q=1-p\) for taking values \(+1\) and -1, respectively, and with mean \(\mu =p-q\). For a fixed positive real number \(\lambda\), let \(N^+ (N^*)\) be the total number of values of n for which \(S_ n>(\lambda +\mu)n (S_ n\geq (\lambda +\mu)n)\) and let \(L^+ (L^*)\) be the supremum of the values of n for which \(S_ n>(\lambda +\mu)n (S_ n\geq (\lambda +\mu)n)\). Explicit expressions for the exact distributions of \(N^+\), \(N^*\), \(L^+\), \(L^*\) are given in this paper when \(\lambda +\mu =\pm k/(k+2)\) for any nonnegative integer k.