First Hitting Time of a High Level by a Catalytic Branching Walk

In a model of a supercritical catalytic branching random walk (CBRW) on the integers $\mathbb {Z}$, the case of light tails of the walk jump is considered, i.e., the Cramér condition is imposed. A limit theorem in the sense of almost sure convergence is proved for the first time of hitting a linearly growing (in time) high level by particles. In the limit, there arises the same constant as in the limit theorem for the maximum of a CBRW.