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

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Summable conditional projections give a stationary martingale CLT

Statement

Assume AC. Let $T$ be an invertible bimeasurable probability-preserving transformation on $(Ω, A, Q)$ such that $Q (A △ T^{- 1} A) = 0$ implies $Q (A) \in {0, 1}$ . Let $X_{0} \in L^{2} (Q)$ be real and centered, put $X_{k} = X_{0} \circ T^{k}$ for $k \in Z$ , and put $F_{k} = σ (X_{i} : i \leq k)$ . Suppose $\sum_{j = 1}^{\infty} ∥ E [X_{j} ∣ F_{0}] ∥_{2} < \infty .$ Then the $L^{2}$ series $H_{0} = \sum_{j \geq 1} E [X_{j} ∣ F_{0}]$ converges. Put $H_{k} = H_{0} \circ T^{k}$ and $D_{k} = X_{k} + H_{k} - H_{k - 1}$ . The $D_{k}$ are stationary square-integrable martingale differences for $F_{k}$ , and for $n \geq 1$ , $S_{n} : = \sum_{k = 1}^{n} X_{k} = \sum_{k = 1}^{n} D_{k} + H_{0} - H_{n}, ∥ H_{0} - H_{n} ∥_{2} \leq 2 ∥ H_{0} ∥_{2} .$ With $τ^{2} = E [D_{1}^{2}]$ , $S_{n} / \sqrt{n} \Rightarrow N (0, τ^{2}),$ including $τ^{2} = 0$ . If additionally $\sum_{j \geq 1} ∣ Cov (X_{0}, X_{j}) ∣ < \infty$ , then $τ^{2} = E [X_{0}^{2}] + 2 \sum_{j = 1}^{\infty} Cov (X_{0}, X_{j}) .$

Facts & Assumptions

Given: AC, the invertible probability-preserving ergodic transformation $T$ , centered real $X_{0} \in L^{2} (Q)$ , its stationary translates, natural past filtration, and the displayed summability assumption.

[A1]

AC supplies the inherited choices for the following conditional-expectation and limit suppliers. (The Axiom of Choice)

[F1]

Conditional expectation is linear, preserves expectation and constants, and contracts the real $L^{2}$ norm. (Basic algebra and order properties of conditional expectation, Conditional lp contraction)

[F2]

Conditional expectations satisfy the tower property and allow a known measurable factor to be taken out when the input and the required products are integrable. (Tower property of conditional expectation, Taking out what is known)

[F3]

Under countable choice $L^{2}$ is complete. (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ )

[F4]

Square-integrable products satisfy Cauchy–Schwarz. (Cauchy-Schwarz for random variables)

[F5]

A stationary integrable process with ergodic canonical shift has averages converging almost surely to its expectation. Ergodicity requires invariant events to have probability zero or one. (Birkhoff limit for a stationary integrable process, Ergodicity relative to an invariant measure)

[F6]

A square-integrable martingale-difference array with terminal conditional variance clock tending to $1$ in probability and vanishing unconditional Lindeberg sums has terminal sum converging to $N (0, 1)$ . (Martingale central limit theorem)

[F7]

Dominated convergence passes limits through integrals dominated by an integrable function. (Dominated convergence)

[F8]

$L^{2}$ convergence implies convergence in probability; a perturbation tending to zero in probability preserves a weak limit; continuous scaling preserves weak convergence. ( $L^{p}$ convergence implies convergence in probability, Converging together lemma, Continuous mapping theorem)

[F9]

Variance and covariance are centered second and product moments. (Moments, variance, and covariance on a probability space)

Proof

1.1A1F1given

For an integrable $Z$ and any integer $k$ , the variable $E [Z ∣ F_{0}] \circ T^{k}$ is $F_{k}$ measurable because the pullback of $F_{0}$ by $T^{k}$ is $F_{k}$ . Every event in $F_{k}$ is the pullback of an event in $F_{0}$ , since $T$ is invertible and bimeasurable. Changing variables under the probability-preserving bijection $T^{k}$ proves that the integrals of this candidate and $Z \circ T^{k}$ agree over every $F_{k}$ event. Uniqueness of conditional expectation therefore gives $E [Z \circ T^{k} ∣ F_{k}] = E [Z ∣ F_{0}] \circ T^{k}$ as almost-everywhere classes.

2.1A1F3step 1.1givenconstruct

By summability and [F3], the partial sums defining $H_{0}$ are Cauchy in $L^{2} (F_{0})$ and have a limit there. Composition with $T^{k}$ is an $L^{2}$ isometry, so step 1.1 yields $H_{k} = \sum_{j \geq 1} E [X_{k + j} ∣ F_{k}]$ in $L^{2}$ , with $H_{k}$ measurable in $F_{k}$ and $∥ H_{k} ∥_{2} = ∥ H_{0} ∥_{2}$ . Representatives may be fixed for the countably many indices; discard a countable union of null sets for their identities.

3.1F1F2F8step 1.1step 2.1algebra

By [F1], conditional expectation is continuous in $L^{2}$ . Apply [F2]'s tower property to the partial sums in step 2.1 and pass to the $L^{2}$ limit to obtain $E [H_{k} ∣ F_{k - 1}] = \sum_{j \geq 1} E [X_{k + j} ∣ F_{k - 1}] = H_{k - 1} - E [X_{k} ∣ F_{k - 1}]$ . Thus $D_{k} = X_{k} + H_{k} - H_{k - 1}$ is square-integrable and $F_{k}$ measurable, with $E [D_{k} ∣ F_{k - 1}] = 0$ . Also $D_{k} = D_{1} \circ T^{k - 1}$ , so the differences are stationary. Summing their defining identity gives $S_{n} = M_{n} + R_{n}$ , where $M_{n} = \sum_{k = 1}^{n} D_{k}$ , $R_{n} = H_{0} - H_{n}$ , and $∥ R_{n} ∥_{2} \leq 2 ∥ H_{0} ∥_{2}$ . In particular $R_{n} / \sqrt{n} \to 0$ in $L^{2}$ and hence in probability by [F8].

4.1F1F5step 1.1step 3.1given

Put $q_{0} = E [D_{1}^{2} ∣ F_{0}]$ . It is nonnegative, integrable, and has expectation $τ^{2} = E [D_{1}^{2}]$ by [F1]. Step 1.1 gives $E [D_{k}^{2} ∣ F_{k - 1}] = q_{0} \circ T^{k - 1}$ . The map $ω \mapsto (q_{0} (T^{j} ω))_{j \geq 0}$ intertwines $T$ with the canonical one-sided shift. Hence a shift-invariant event for this process pulls back to a $T$ -invariant event modulo null sets, whose probability is zero or one by the assumed ergodicity. The process is stationary, integrable and ergodic, so [F5] gives $n^{- 1} \sum_{k = 1}^{n} E [D_{k}^{2} ∣ F_{k - 1}] \to τ^{2}$ almost surely. The limiting variance is therefore a constant.

5.1A1F6F7F8step 3.1step 4.1construct

If $τ^{2} > 0$ , let $Z_{n, k} = D_{k} / (τ \sqrt{n})$ for $1 \leq k \leq n$ , and let $Z_{n, k} = 0$ for $k > n$ ; use the filtration $F_{n, k} = F_{\min (k, n)}$ for $k \geq 0$ . Step 3.1 makes each row a square-integrable martingale-difference row. The row sums and clocks are eventually constant, and step 4.1 shows their terminal clocks converge almost surely to $1$ . For every $ε > 0$ , stationarity gives $\sum_{k \geq 1} E [Z_{n, k}^{2} 1_{{∣ Z_{n, k} ∣ > ε}}] = τ^{- 2} E [D_{1}^{2} 1_{{∣ D_{1} ∣ > ε τ \sqrt{n}}}] \to 0$ by [F7]. Thus [F6] gives $M_{n} / (τ \sqrt{n}) \Rightarrow N (0, 1)$ . Scaling and the perturbation in step 3.1 give $S_{n} / \sqrt{n} \Rightarrow N (0, τ^{2})$ by [F8].

6.1F8step 3.1step 5.1given

If $τ^{2} = 0$ , stationarity implies $E [D_{k}^{2}] = 0$ for every $k$ , so $M_{n} = 0$ almost surely. Step 3.1 gives $S_{n} / \sqrt{n} \to 0$ in $L^{2}$ and hence in probability. The constant sequence $0$ has weak limit the point mass $N (0, 0)$ ; [F8] transfers that limit to $S_{n} / \sqrt{n}$ . This establishes the asserted CLT in both variance cases.

7.1F2F4F7F9step 3.1given∎

Suppose the additional covariance summability holds, and put $γ_{j} = Cov (X_{0}, X_{j})$ . Stationarity and expansion of the finite square give $Var (S_{n}) / n = γ_{0} + 2 \sum_{j = 1}^{n - 1} (1 - j / n) γ_{j} \to γ_{0} + 2 \sum_{j \geq 1} γ_{j}$ , by [F7] on counting measure with summable dominator $∣ γ_{j} ∣$ . For $i < j$ , $D_{i}$ is $F_{j - 1}$ measurable, so [F2], [F4], and step 3.1 give $E [D_{i} D_{j}] = 0$ . Therefore $E [M_{n}^{2}] = n τ^{2}$ . The identity $S_{n} = M_{n} + R_{n}$ and [F4] imply $∣ E [S_{n}^{2}] - n τ^{2} ∣ / n \leq ∥ R_{n} ∥_{2}^{2} / n + 2 ∥ R_{n} ∥_{2} τ / \sqrt{n} \to 0$ , since $∥ R_{n} ∥_{2}$ is bounded. All $S_{n}$ are centered, so [F9] identifies the two variance limits and proves the stated covariance formula, including $τ^{2} = 0$ .

Source notes

The absolutely summable projection assumption permits an exact $L^{2}$ coboundary series, so no maximal martingale approximation theorem is imported. The deterministic variance follows from the explicitly assumed ergodicity and its factor argument. The covariance identity is obtained from second moments, independently of convergence in distribution.

Depends on

Used by

A rho-mixing CLT for square-integrable stationary sequences Theorem

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Sources

Mikhail Gordin and Magda Peligrad, On the functional central limit theorem via martingale approximation, §1 and full martingale construction in §2; the absolutely summable specialization is derived locally (standard reference, not scraped)