Suppose I have $n$ (not independent) complex random variables $X_i,i=1,\ldots,n$. I want to show the following \begin{equation} \mathbb{E}\left[\left|\sum_{j=1}^nX_j\right|^2\right]^{1/2}\leq \sum_{j=1}^n\mathbb{E}[|X_j|^2]^{1/2}. \end{equation} I suppose it's somewhat immediate from applying Jensen's inequality in a clever way, but haven't been able to figure it out by myself. Thanks for your help!
2026-03-26 02:52:04.1774493524
Show basic inequality for complex random variables
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