WEAK-CONVERGENCE OF SUMS OF MOVING AVERAGES IN THE ALPHA-STABLE DOMAIN OF ATTRACTION

Skorohod has shown that the convergence of sums of i.i.d. random variables to an alpha-stable Levy motion, with 0 < alpha < 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 sup and inf functionals to be continuous.