The paper investigates a random walk \(S_n:=\sum_{k\geq 1} X_k\) (\(S_0:=0\)) governed by probabilities \(P\{X_k=1|S_{i-1}=2m\}=\alpha\), \(P\{X_k=-1|S_{i-1}=2m\}=1-\alpha\), \(P\{X_k=1|S_{i-1}=2m+1\}=1-\alpha\) and \(P\{X_k=-1|S_{i-1}=2m+1\}=\alpha\). If \(\alpha=1/2\), \((S_n)\) reduces to the simple symmetric random walk which is a fairly classical object.
By applying multivariate Lagrange inversion to the corresponding generating functions, the exact distributions of the first-passage time \(T_a:=\inf\{n: S_n=a\}\) and the maximum \(M_n:=\underset{0\leq i\leq n}{\max}\,S_i\) are obtained. With the aid of the singularity analysis of generating functions the asymptotics of \(P\{T_a=n\}\) and \(EM_n\), as \(n\to\infty\), are found, and the weak convergence of \(M_n/\sqrt{n}\) was proved. Further, by using combinatorial arguments formulae for \(P\{S_n=k\}\) are established in three situations: (a) there is additionally one absorbing boundary; (b) there are additionally two absorbing boundaries; (c) there are no absorbing boundaries. The paper closes with applications of the results obtained to (1) a generalization of the classical coin-tossing; (2) the busy period of a chemical chain.
Reviewer’s remark: On p. 175, the authors claim that the limiting distribution of \(M_n/\sqrt{n}\) is stable with index \(1/2\). This contradicts to their formula (30) which asserts that the limiting distribution is the law of \(|N|\), where \(N\) is a random variable with normal distribution.