# The structure of self-similar stable mixed moving averages

@article{Pipiras2002TheSO,
title={The structure of self-similar stable mixed moving averages},
journal={Annals of Probability},
year={2002},
volume={30},
pages={898-932}
}
• Published 1 April 2002
• Mathematics
• Annals of Probability
Let fi2 (1;2) and Xfi be a symmetric fi-stable (SfiS) process with stationary increments given by the mixed moving average Xfi(t) = Z
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## References

SHOWING 1-10 OF 35 REFERENCES
(1/a)-self similar a-stable processes with stationary increments
• Mathematics
• 1990
In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the [alpha]-stable Levy motion is the only (1/[alpha])-self-similar [alpha]-stable process with stationary
A remark on self‐similar processes with stationary increments
The upper bound of the parameter of self-similar processes with stationary increments is given in terms of the moment condition. On calcule le majorant du parametre de processus autosimilaires
On uniqueness of the spectral representation of stable processes
In this paper we show that any two spectral representations of a symmetric stable process may differ only by a change of variable and a parameter-independent multiplier. Our result can immediately be
Log-fractional stable processes
• Mathematics
• 1988
The first problem attacked in this paper is answering the question whether all 1/[alpha]-self-similar [alpha]-stable processes with stationary increments are [alpha]-stable motions. The answer is yes
Characterization of linear and harmonizable fractional stable motions
• Mathematics
• 1992
We characterize the linear and harmonizable fractional stable motions as the self-similar stable processes with stationary increments whose left-equivalent (or right-equivalent) stationary processes
Spectral representation and structure of self-similar processes
• Mathematics
• 1997
In this paper we establish a spectral representation of any symmetric stable self-similar process in terms of multiplicative flows and cocycles. Applying the Lamperti transformation we obtain a
Stable mixed moving averages
• Mathematics
• 1993
SummaryThe class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable
Structure of stationary stable processes
A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique
Decomposition of self-similar stable mixed moving averages
• Mathematics
• 2002
Abstract. Let α? (1,2) and Xα 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
Using Renewal Processes to Generate Long-Range Dependence and High Variability
• Mathematics
• 1986
We explore here three types of convergence theorems involving the normalized partial sums of two random processes W = W(t) and V = V(t) indexed by the integers t = ...,−1, 0.1,... . W(t) is a