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

## TL;DR

This paper studies the convergence and geometric properties of a $p$-stable fractional wavelet series, revealing new insights into its behavior and self-similarity in the context of wavelet analysis and fractional operators.

## Contribution

It introduces conditions for convergence of $p$-stable wavelet series involving fractional integrals and explores their geometric and self-similar properties.

## Key findings

- Series converges under specific conditions involving $p$-stability and fractional integrals.
- Identifies geometric properties related to self-similarity of the series.
- Extends analysis to modified fractional integral operators.

## 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.