Left-continuous random walk on Z and the parity of its hitting times

When it comes to random walk on the integers Z, the arguably first step of generalization beyond simple random walk is the class of one-sidedly continuous random walk, where the stepsize in only one direction is bounded by 1. Moreover, the time until state 0 is hit by left-continuous random walk on Z has a direct connection to the total progeny in branching processes. In this article, the probability of left-continuous random walk to be negative at an even (resp. \ odd) time is derived and used to determine the probability of nearly left-continuous random walk to eventually become negative.