Gabor frames generated by Random-Periodic time-frequency shifts

In this article, we consider a variation of the existence of Gabor frames in a probabilistic setting, in which we consider time-frequency shifts taken over random-periodic sets. We demonstrate that the method of selecting random-periodic time-frequency shifts is successful with high probability for specific categories of well-behaved functions, notably including Hermite functions, totally positive functions, and B-spline functions. In particular, we show that if $x_1, x_2, \ldots ,x_m$ are independent and uniformly distributed in $[0,1),$ with $m$ sufficiently large, then the set of time-frequency shifts $\Lambda \times \ZZ,$ where $\Lambda=\ZZ + \{x_1, x_2, \ldots, x_m\},$ forms Gabor frame with high probability.