Convergence of $p$-Stable Random Fractional Wavelet Series and Some of its Properties

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