Taylor coefficients condition and mean growth of random series in $Q_K$ spaces

Some properties for random power series $(\mathcal{R}f)(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{ϵ}_n{z}^n$ in $Q_K$ spaces and $Q_{K,0}$ spaces were studied in this paper. It was proved that if the condition $\sum _{n=1}^{\mathrm{\infty }}n|a_n{|}^{2}K(\frac{1}{n})<\mathrm{\infty }$ holds, then $\mathcal{R}f\in Q_K$ a.s. Furthermore, this condition actually implied that $\mathcal{R}f\in Q_{K,0}$ a.s. As an application, the relationship between mean Lipschitz spaces and $Q_K$ spaces was shown.