# The structure of self-similar stable mixed moving averages

@article{Pipiras2002TheSO, title={The structure of self-similar stable mixed moving averages}, author={Vladas Pipiras and Murad S. Taqqu}, journal={Annals of Probability}, year={2002}, volume={30}, pages={898-932} }

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

## 31 Citations

Decomposability for stable processes

- Mathematics
- 2011

We characterize all possible independent symmetric α-stable (SαS) components of an SαS process, 0<α<2. In particular, we focus on stationary SαS processes and their independent stationary SαS…

Random rewards, fractional Brownian local times and stable self-similar processes

- Mathematics
- 2006

We describe a new class of self-similar symmetric $\alpha$-stable processes with stationary increments arising as a large time scale limit in a situation where many users are earning random rewards…

Mixed Moving Averages and Self-Similarity

- Mathematics
- 2017

The focus of the chapter is on a large class of symmetric stable self-similar processes with stationary increments, known as self-similar mixed moving averages. Minimal representations of…

Decomposition of discrete time periodically correlated and multivariate stationary symmetric stable processes

- Mathematics
- 2005

The spectral structure of discrete time periodically correlated (as well as multivariate stationary) symmetric [alpha]-stable processes is identified by decomposing such a process uniquely in…

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…

Long-Range Dependence as a Phase Transition

- Physics
- 2016

Long-range dependence in a stationary process has been understood as corresponding to a particular second-order behavior, to a particular range of the Hurst parameter, or of fractional integration.

Integral representations of periodic and cyclic fractional stable motions

- Mathematics
- 2004

Stable non-Gaussian self-similar mixed moving averages can be decomposed into several components. Two of these are the periodic and cyclic fractional stable motions which are the subject of this…

Scaling Properties of the Empirical Structure Function of Linear Fractional Stable Motion and Estimation of Its Parameters

- Mathematics
- 2015

Linear fractional stable motion is an example of a self-similar stationary increments stochastic process exhibiting both long-range dependence and heavy-tails. In this paper we propose methods that…

Random-Time Isotropic Fractional Stable Fields

- Mathematics
- 2011

Generalizing both Substable Fractional Stable Motions (FSMs) and Indicator FSMs, we introduce α-stabilized subordination, a procedure which produces new FSMs (H-self-similar, stationary increment…

SEMI-ADDITIVE FUNCTIONALS AND COCYCLES IN THE CONTEXT OF SELF-SIMILARITY

- Mathematics
- 2004

Kernel functions of stable, self-similar mixed moving averages are known to be related to nonsingular flows. We identify and examine here a new functional occuring in this relation and study its…

## 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

- Mathematics
- 1986

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

- Mathematics
- 1994

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

- Mathematics
- 1995

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…