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.