Occupation probability of a correlated random walk and a correlated ruin problem (Q1819472)

The authors study a random walk \(S_ n=X_ 0+X_ 1+...+X_ n\), where \(X_ i=+1\) or -1 for \(i>0\) and for some \(k\geq 0\) the sequence \((X_ n,X_{n+1},...,X_{n+k})\), \(n>0\), forms a Markov chain. They calculate in the case \(k=0\) the probabilities \(P(S_ n=j)\) and investigate ruin probabilities in the cases \(k\leq 2\).