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

Abstract

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

19 pages, 2 figures