  For appropriate orthonormal wavelet basis $\{\psi_{j\,k}^e
\}_{j\in\mathbb{Z}\,k\in\mathbb{Z}^d\,e\in\{0,1\}^d}$, constants $p$ and
$\gamma$, if $\mathcal{I}_{\gamma}$ denotes the Riesz fractional integral
operator of order $\gamma$ and $(\eta_{j\,k\,e})_{j\in\mathbb{Z}
k\in\mathbb{Z}^d \,e\in\{0,1\}^d}$ a sequence of independent identically
distributed symmetric $p$-stable random variables, we investigate the
convergence of the series $\sum\limits_{j\,k\,e} \eta_{j\,k\,e}
\mathcal{I}_{\gamma} \psi_{j\,k\,}^e$. Similar results are also studied for
modified fractional integral operators. Finally, some geometric properties
related to self similarity are studied.
