showing this bound

Question

Let $a\in\mathbb{R}^N$ and $X_1,...,X_N$ be independent random variables with zero mean and unit variance. Im trying to prove that:

$$\|a\|_{2}\le{\|\sum_{i=1}^{N}{a_iX_i}}\|_{L^p},$$ with $p\in{[2,\infty)}$

Ive tried playing with definitions but haven't gotten anywhere.

Any hints?

Answer 1

Let $Y=\displaystyle \sum_{i=1}^N a_i X_i$. By the hypothesis($X_i$ 's independent, with mean zero and unit variance) we can easily find that $||Y||_{L^2}:= (E|Y| ^2)^{1/2}=(\displaystyle \sum_{i=1}^N a_i^2)^{1/2}=||a||_{L^2}$. Since $p\geq 2$ it follows from Lyapunov inequality. See Lyapunov's inequality in Probability