SCHWEITZER 1977 Problem 10

Let the sequence of random variables $\{ X_m, m \geq 0\ \}, X_0=0$ , be an infinite random walk on the set of nonnegative integers with transition probabilities $$p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, i \geq 0 \,$$ $$q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, i>0.$$ Prove that for arbitrary $k >0$ there is an $\alpha_k > 1$ such that $$P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)$$ satisfies the limit relation $$\lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.$$

J. Tomko