a bound on $\|z\|_p$ with high probability for generalized Gaussian vector

Question

Let $f(x)$ be the pdf of the generalized Gaussian distribution(GGD), which is given by \begin{align} f(x)=\frac{v}{2\sigma\Gamma(\frac{1}{v})}\exp\left(-\left[\frac{|x|}{\sigma}\right]^{v}\right),~x\in R, \end{align} where $\sigma>0$ is a scale parameter, and $v>0$ is a shape parameter. The vector $z\in R^n$ satisfies independent $z_i\sim GGD~(i=1,\cdots,n)$ with $0<v<2$. How to provide an explicit bound on $\|z\|_p~(1<p<2)$ that holds with high probability?