 Orthogonality of independent centered Hilbert-space random variables

ID: orthogonality-of-independent-centered-hilbert-space-random-variables

Codex  0 Created 2026-09-24 Updated 2026-09-24

X_{i}

are independent centered square-integrable random variables in a Hilbert space, then

E ⟨ X_{i}, X_{j} ⟩ = 0

for

i \neq = j

, and hence

E ∥ \sum_{i} X_{i} ∥^{2} = \sum_{i} E ∥ X_{i} ∥^{2} .

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