Orthogonality of independent centered Hilbert-space random variables
= Orthogonality of independent centered Hilbert-space random variables
If $X_i$ are independent centered square-integrable random variables in a Hilbert space, then $\mathbb E\langle X_i,X_j\rangle=0$ for $i\ne j$, and hence
$$
\mathbb E\left\lVert\sum_iX_i\right\rVert^2
=\sum_i\mathbb E\lVert X_i\rVert^2.
$$
= Independent random variable
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