LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Independent centered Hilbert-valued elements have zero expected inner product

Statement

Let $H$ be a real Hilbert space with a fixed complete finite or countable orthonormal basis $(e_{k})$ . On a probability space let $U, V$ be independent, strongly measurable and Bochner-integrable $H$ -valued random elements with $E_{B} U = E_{B} V = 0$ and $E ∥ U ∥_{H}^{2}, E ∥ V ∥_{H}^{2} < \infty$ . Then $E ⟨ U, V ⟩_{H} = 0.$

Facts & Assumptions

Given: The hypotheses and fixed basis in the Statement.

[F1]

Bochner integrability makes the means in the Statement well-defined, and a bounded linear functional commutes with the Bochner integral; in particular $E ⟨ U, e_{k} ⟩_{H} = ⟨ E_{B} U, e_{k} ⟩_{H} = 0$ , and similarly for $V$ (Bochner-integrable function, Bounded linear maps commute with Bochner integration).

[F2]

Measurable scalar functions of independent random elements are independent (Independent random elements, Measurable coordinatewise functions preserve independence).

[F3]

Expectations of integrable products of independent real random variables factor (Expectations factor over finite products of independent random variables).

[F4]

For each vector, the finite orthogonal projections onto the initial basis spans converge in norm to that vector and are contractions (Complete orthonormal bases give norm-convergent projections and Parseval).

[F5]

Cauchy--Schwarz makes $∥ U ∥_{H} ∥ V ∥_{H}$ integrable, and dominated convergence passes expectations through the projection limit (Cauchy-Schwarz for random variables, Dominated convergence).

Proof

Given: The independent centered random elements and basis in the Statement.

1.1F1F2F3algebra

For each $k$ , the coordinate variables $U_{k} = ⟨ U, e_{k} ⟩_{H}$ and $V_{k} = ⟨ V, e_{k} ⟩_{H}$ are measurable, centered by [F1], and independent by [F2]. Their product is integrable by Cauchy--Schwarz, so [F3] gives $E [U_{k} V_{k}] = E [U_{k}] E [V_{k}] = 0$ . Every finite sum $⟨ P_{N} U, P_{N} V ⟩_{H} = \sum_{k \leq N} U_{k} V_{k}$ therefore has expectation zero.

2.1F4F5step 1.1algebra∎

By [F4], $P_{N} U \to U$ and $P_{N} V \to V$ almost surely (indeed for every vector in $H$ ); continuity of the inner product gives $⟨ P_{N} U, P_{N} V ⟩_{H} \to ⟨ U, V ⟩_{H}$ . Since projections are contractions, $∣ ⟨ P_{N} U, P_{N} V ⟩_{H} ∣ \leq ∥ U ∥_{H} ∥ V ∥_{H},$ and this majorant is integrable by [F5]. Dominated convergence [F5] and step 1.1 now yield $E ⟨ U, V ⟩_{H} = 0$ .

Depends on

Used by

Hilbert-valued CLT for L2-m-approximable Bernoulli shifts Theorem

Dependency tree · two levels

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Sources

Gerald Teschl, Topics in Real and Functional Analysis, §2.1, Theorems 2.2–2.4, printed pp.49–51 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 2.1.12 and proof, §2.1.2, printed pp.48–49 (standard reference, not scraped)