The structure of self-similar stable mixed moving averages

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.