Left-Continuous Random Walk on $${\mathbb {Z}}$$ and the Parity of Its Hitting Times

Abstract When it comes to random walk on the integers $${\mathbb {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> , 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 $${\mathbb {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> started at 1 has a direct connection to the total progeny in branching processes. In the analysis of Maker–Breaker games on trees arising from these branching processes, however, the corresponding random walks have increments bounded from below by $$-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> instead of $$-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In this article, the probability of left-continuous random walk started at 0 to be negative at an even (resp. odd) time is derived and used to determine the probability of such nearly left-continuous random walk started at 0 to eventually become negative.