There is no test to prove a distribution is non-normal stable. However there are tests that indicate stability. One of these is a test for infinite variance. For the normal (a special case of stable) distribution the variance converges to a finite real number as grows without bounds. When tails are heavy (stable ) variance does not exist or is infinite. Granger and Orr (1972) devised a running variance test for infinite variance that is displayed here.
Note that when , the distribution is normal and the plot of the test shows the variance converging. At lower levels of the plot remains "wild" indicating infinite or nonexistent variance.