Random Power Series in Q p , 0 Spaces

Abstract

Aulaskari et al. proved if $0<p<1$ and $\left\{{\epsilon }_n\right\}$ is sequence of independent, identically distributed Rademacher random variables on a probability space, then the condition ${\Sigma }_{n=0}^{\infty }{n}^{1-p}{\left|a_n\right|}^2<\infty$ implies that the random power series $\left(\mathcal{R}f\right)\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{\epsilon }_n{z}^n\in Q_p$ almost surely. In this paper, we improve this result showing that the condition ${\Sigma }_{n=0}^{\infty }{n}^{1-p}{\left|a_n\right|}^2<\infty$ actually implies $\mathcal{R}f\in Q_{p,0}$ almost surely.