The Annals of Probability

The structure of self-similar stable mixed moving averages

Vladas Pipiras and Murad S. Taqqu

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Abstract

Let $\alpha \in (1,2)$ and $X_{\alpha}$ be a symmetric $\alpha$-stable $(S \alpha S)$ process with stationary increments given by the mixed moving average $$X_{\alpha}(t) = \int_X \int_{\mathbb{R}}(G(x, t + u) - G(x, u)) M_{\alpha}(dx, du), \quad t \in \mathbb{R},$$ where $(X, \mathscr{X}, \mu)$ is a standard Lebesgue space, $G : X \times \mathbb{R} \mapsto \mathbb{R}$ is some measurable function and $M_{\alpha}$ is a $(S \alpha S)$ random measure on $X \times \mathbb{R}$ with the control measure $m(dx, du) = \mu (dx) du$. Assume, in addition, that $X_{\alpha}$ is self-similar with exponent $H \in (0,1)$. In this work, we obtain a unique in distribution decomposition of a process $X_{\alpha}$ into three independent processes

$$X_{\alpha} =^d X_{\alpha}^{(1)} + X_{\alpha}^{(2)} + X_{\alpha}^{(3)}.$$

We characterize $X_{\alpha}^{(1)}$ and $X_{\alpha}^{(2)}$ and provide examples of $X_{\alpha}^{(3)}$.

The first process $X_{\alpha}^{(1)}$ can be represented as

$$\int_Y \int_{\mathbb{R}} \int_{\mathbb{R}} e^{\kappa s}(F(y, e^s (t + u)) - F(y, e^s u)) M_{\alpha} (dy, ds, du),$$

where $\kappa = H - \frac{1}{\alpha}, (Y, \mathscr{Y}, \nu)$ is a standard Lebesgue space and $M_{\alpha}$ is a $(S \alpha S)$ random measure on $Y \times \mathbb{R} \times \mathbb{R}$ with the control measure $m(dy, ds, du) = \nu(dy) ds du$. Particular cases include the limit of renewal reward processes, the so-called "random wavelet expansion" and Takenaka process. The second process $X_{\alpha}^{(2)}$ has the representation

$$\int_Z \int_{\mathbb{R}} (G_1(z)((t + u)_+^{\kappa} - u_+^{\kappa}) + G_2 (z)((t + u)_-^{\kappa} - u_-^{\kappa})) M_{\alpha}(dz, du), \quad \text{if $\kappa \not= 0$},$$

$$\int_Z \int_{\mathbb{R}} (G_1(z)(\ln |t + u| - \ln |u|) + G_2 (z)(1_{(0, \infty)}(t + u) - 1_{(0, \infty)}(u))) M_{\alpha}(dz, du), \quad \text{if $\kappa = 0$},$$

where $Z, \mathscr{Z}, v)$ is a standard Lebesgue space and $M_{\alpha}$ is a $S \alpha S$ random measure on $Z \times \mathbb{R}$ with the control measure $m(dz, du) = v(dz)du$. Particular cases include linear fractional stable motions, log-fractional stable motion and $S \alpha S$ Lévy motion. And example of a process $X_{\alpha}^{(3)}$ is $$\int_0^1 \int_{\mathbb{R}} ((t + u)_+^{\kappa} 1_{[0, 1/2)} ({x + \ln |t + u|}) - u_+^{\kappa} 1_{[0, 1/2) ({x + \ln |u|})) M_{\alpha} (dx, du),$$ where $M_{\alpha}$ is a $S \alpha S$ random measure on $[0, 1) \times \mathbb{R}$ with the control measure $m(dx, du) = dxdu$ and ${\cdot}$ is the fractional part function.

Article information

Source
Ann. Probab., Volume 30, Number 2 (2002), 898-932.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481011

Digital Object Identifier
doi:10.1214/aop/1023481011

Mathematical Reviews number (MathSciNet)
MR1905860

Zentralblatt MATH identifier
1016.60057

Subjects
Primary: 60G18: Self-similar processes 60G52: Stable processes
Secondary: 28D 37A

Keywords
stable self-similar processes with stationary increments dissipative and conservative flows cocycles semi-additive functionals

Citation

Pipiras, Vladas; Taqqu, Murad S. The structure of self-similar stable mixed moving averages. Ann. Probab. 30 (2002), no. 2, 898--932. doi:10.1214/aop/1023481011. https://projecteuclid.org/euclid.aop/1023481011


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References

  • BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • BURNECKI, K., ROSI ´NSKI, J. and WERON, A. (1998). Spectral representation and structure of stable self-similar processes. In Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943-1995 (I. Karatzas, B. S. Rajput and M. S. Taqqu, eds) 1-14. Birkhäuser, Boston.
  • CAMBANIS, S., MAEJIMA, M. and SAMORODNITSKY, G. (1992). Characterization of linear and harmonizable fractional stable motions. Stochastic Processes Appl. 42 91-110.
  • CHI,(2001). Construction of stationary self-similar generalized fields by random wavelet expansion. Probab. Theory Related Fields 121 269-300.
  • KASAHARA, Y., MAEJIMA, M. and VERVAAT, W. (1988). Log-fractional stable processes. Stochastic Processes Appl. 30 329-339.
  • KRENGEL, U. (1969). Darstellungssätze für Strömungen und Halbströmungen II. Math. Ann. 182 1-39.
  • KRENGEL, U. (1985). Ergodic Theorems. De Gruyter, Berlin.
  • PIPIRAS, V. and TAQQU, M. S. (2000). The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6 607-614.
  • PIPIRAS, V. and TAQQU, M. S. (2001a). Decomposition of self-similar stable mixed moving averages. Probab. Theory Related Fields. To appear.
  • PIPIRAS, V. and TAQQU, M. S. (2001b). Dilated fractional stable motions. Preprint.
  • ROSI ´NSKI, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163-1187.
  • ROSI ´NSKI, J. (1998). Minimal integral representations of stable processes. Preprint.
  • SAMORODNITSKY, G. and TAQQU, M. S. (1990). (1/)-self-similar processes with stationary increments. J. Multivariate Anal. 35 308-313.
  • SAMORODNITSKY, G. and TAQQU, M. S. (1994). Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York.
  • SURGAILIS, D., ROSI ´NSKI, J., MANDREKAR, V. and CAMBANIS, S. (1993). Stable generalized moving averages. Probab. Theory Related Fields 97 543-558.
  • TAKENAKA, S. (1991). Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 1-12.
  • BOSTON, MASSACHUSETTS 02215 E-MAIL: pipiras@math.bu.edu murad@math.bu.edu