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Decomposition of self-similar stable mixed moving averages
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Decomposition of self-similar stable mixed moving averages
Vladas Pipiras 1 and Murad S. Taqqu1
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Department of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215, USA. e-mail: pipiras@math.bu.edu, murad@math.bu.edu, US |
Abstract. Let α? (1,2) and X
α
be a symmetric α-stable (S α S) process with stationary increments given by the mixed moving average
where is a standard Lebesgue space, is some measurable function and M
α
is a SαS random measure on X ×ℝ with the control measure m
α
(dx, du) = μ(dx)du. We show that if X
α
is self-similar, then it is determined by a nonsingular flow, a related cocycle and a semi-additive functional. By using
the Hopf decomposition of the flow into its dissipative and conservative components, we establish a unique decomposition in
distribution of X
α
into two independent processes
where the process X
α
D
is determined by a nonsingular dissipative flow and the process X
α
C
is determined by a nonsingular conservative flow. In this decomposition, the linear fractional stable motion, for example,
is determined by a conservative flow.
Received: 20 June 2000 / Revised version: 6 September 2001 / Published online: 14 June 2002
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