Weak convergence of sums of moving averages in the α-stable domain of attraction
Taqqu, Murad S.
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Citation (published version)F. Avram, M.S. Taqqu. 1992. "Weak Convergence of Sums of Moving Averages in the α -Stable Domain of Attraction." The Annals of Probability, Volume 20, Issue 1, pp. 483 - 503 (21). https://doi.org/10.1214/aop/1176989938
Skorohod has shown that the convergence of sums of i.i.d. random variables to an a-stable Levy motion, with 0 < a < 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 conver- gence 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 con- vergence. M1 is weaker than J1, but it is strong enough for the sup and inf functionals to be continuous.
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