Multivariate Lagrange inversion and the maximum of a persistent random walk
Let \(\alpha \in (0,1)\) and let \(X_1, X_2, \ldots\) be the time-homogeneous Markov chain with values in \(\{+1, -1\}\) and transition probabilities given by \[ P(X_i =1\mid X_{i-1} =1) = P(X_i =-1\mid X_{i-1} =-1) = \alpha, \] \[ P(X_i =1\mid X_{i-1} =-1) = P(X_i =-1\mid X_{i-1} =1) = 1-\alpha. \] The persistent random walk \((S_n)\) is then given by \(S_n = \sum_{i=1}^n X_i\). Using multivariate Lagrange inversion, the author obtains a formula for the absorption times of this random walk which has also been derived by Mohanty (1979). He uses this result to determine the asymptotic distribution of the maximum of the persistent random walk.
scientific article; zbMATH DE number 1741587