Large jumps of $q$-Ornstein-Uhlenbeck processes

We continue the investigation of sample paths of $q$-Ornstein-Uhlenbeck process. We show that for all $q\in (-1,1)$, the process has big jumps crossing from near one end point of the domain to the other with positive probability. Moreover, the number of such jumps in an appropriately enlarged window converges weakly to a Poisson random variable.