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

Abstract

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.