I am trying to prove the upper bound for exercise 2.6.5 in the following notes: http://home.ustc.edu.cn/~liweiyu/documents/HDP-book.pdf
I need to show that: $$ \| \sum_{i=1}^N a_i X_i \|_{L^p} \leq C K \sqrt{p} \| a \|_{2}^2 $$
where $p \geq 2$, $X_i$ are independent sub-gaussian RV's with mean zero and unit variance, $C>0$ is an absolute constant and $K = \text{max}_{1 \leq i \leq n} ( \| X_i \|_{\psi_2} )$
So far I have that by using the General Hoeffing inequality that:
$$ \| \sum_{i=1}^N a_i X_i \|_{L^p} = \int_{0}^{\infty} pt^{p-1} \mathbb{P}( \sum_{i=1}^N a_i X_i \geq t) \mathrm{d}t \leq \int_{0}^{\infty} 2 \text{exp}(-\frac{ct^2}{K^2 \|a\|_{2}^2}) pt^{p-1} \mathrm{d}t $$
However I am unable to get anywhere in order for the final bound to pop out.
Let $S=\sum^N_{n=1}a_nX_n$ is such that $\sum^N_{k=1}a^2_n=1=\|S\|^2_2$. Let $\kappa^2:=c/K^2$. For $p\geq2$ \begin{align} \int |S|^p\,dP&=p\int^\infty_0 \lambda^{p-1}P[|S|>\lambda]\,d\lambda\\ &\leq 2p\int^\infty_0 \lambda^{p-1}e^{-(\kappa\lambda)^2/2}\,d\lambda\\ &=\frac{1}{\kappa^p}2p\int^\infty_0(2t)^{\tfrac{p-2}{2}}e^{-t}\,dt\\ &=\frac{1}{\kappa^p} 2^{p/2+1}\Gamma\big(p/2 +1\big)=:K^p_p \end{align} where we have used the change of variables $t=(\kappa\lambda)^2/2$. This shows that for any finite linear combination $X=\sum^n_{k=1}\alpha_nX_n$ $$\|X\|_p\leq K_p\|X\|_2$$ where $K_p=1$ for $0<p\leq 2$ and $K_p=\Big(\frac{1}{\kappa^p}2^{\tfrac{p}{2}+1}\Gamma(\tfrac{p}{2}+1)\Big)^{1/p}$ for $p>2$.