Open Access
Translator Disclaimer
January, 1992 Weak Convergence of Sums of Moving Averages in the $\alpha$-Stable Domain of Attraction
Florin Avram, Murad S. Taqqu
Ann. Probab. 20(1): 483-503 (January, 1992). DOI: 10.1214/aop/1176989938


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-$J_1$ sense. $J_1$ is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-$J_1$ 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-$M_1$ convergence. $M_1$ is weaker than $J_1$, but it is strong enough for the $\sup$ and $\inf$ functionals to be continuous.


Download Citation

Florin Avram. Murad S. Taqqu. "Weak Convergence of Sums of Moving Averages in the $\alpha$-Stable Domain of Attraction." Ann. Probab. 20 (1) 483 - 503, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0747.60032
MathSciNet: MR1143432
Digital Object Identifier: 10.1214/aop/1176989938

Primary: 60F17
Secondary: 60J30

Keywords: $J_1$ topology , $M_1$ topology , Levy stable motion , moving averages , stable distribution , weak convergence

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 1 • January, 1992
Back to Top