Rootzen (1978) gives a sufficient condition for sample continuity of moving average processes with respect to stable motion with index $\alpha$ less than two. We provide a simple proof of this criterion for $\alpha < 1$ and show that the condition is then also necessary for continuity of the process. The same result holds for the moving-maximum process. A description of the local behaviour of the sample functions of such processes is given.
A. A. Balkema. L. De Haan. "Almost Sure Continuity of Stable Moving Average Processes with Index Less Than One." Ann. Probab. 16 (1) 333 - 343, January, 1988. https://doi.org/10.1214/aop/1176991905