Weak Convergence of Sums of Moving Averages in the α-Stable Domain of Attraction

Skorohod has shown that the convergence of sums of i.i.d. random variables to an α-stable Levy motion, with 0 < α < 2, holds in the weak-J₁ sense. J₁ is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-J₁ convergence cannot hold because adjacent jumps of the process can coalesce in the limit; however, if the moving average coefficients are positive, then the adjacent jumps are essentially monotone and one can have weak-M₁ convergence. M₁ is weaker than J₁, but it is strong enough for the and inf functionals to be continuous.