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Green–Kubo covariance limit under absolute integrability

Statement

Let $(Ξ, F, P)$ be a probability space and let $V : R \times Ξ \to R^{d}$ be jointly measurable for the completed product of Lebesgue measure and $P$ . Assume $E [V (t)] = 0$ for every $t$ , $E ∣ V (0) ∣^{2} < \infty$ , and second-order stationarity in the explicit sense that each covariance below is finite and $C_{i j} (s) : = E [V_{i} (u) V_{j} (u + s)]$ is independent of $u \in R$ for every $s \in R$ and every $1 \leq i, j \leq d$ . For $T > 0$ , the Lebesgue integral $X (T) = \int_{0}^{T} V (s) d s$ exists almost surely and in $L^{2}$ . Define $Cov (X (T))_{i j} = E [(X_{i} (T) - E X_{i} (T)) (X_{j} (T) - E X_{j} (T))];$ the centered hypothesis makes these means zero. Then $Cov (X_{i} (T), X_{j} (T)) = \int_{0}^{T} (T - s) (C_{i j} (s) + C_{j i} (s)) d s .$ If each $C_{i j}$ is absolutely integrable on $[0, \infty)$ , then $\frac{Cov (X (T))}{2 T} ⟶ D, D_{i j} = \frac{1}{2} \int_{0}^{\infty} (C_{i j} (s) + C_{j i} (s)) d s,$ and $D$ is positive semidefinite. This is a variance-growth conclusion; it does not assert a central limit theorem or identify a transport coefficient.

Facts & Assumptions

Given: A probability space, a jointly measurable centered vector process with stationary second moments and finite second moment at one time, plus absolute integrability of every nonnegative-lag covariance entry for the limit clause.

[A1]

${A C}_{ω}$ is assumed for the completed-product Tonelli–Fubini supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Tonelli and Fubini for the completed product, with only almost-everywhere section measurability: on a completed product of sigma-finite spaces, nonnegative integrals and integrable functions may be integrated by sections under its stated ${A C}_{ω}$ hypothesis.

[F2]

Dominated convergence: almost-everywhere convergence under one common integrable majorant allows passage to the limit in the integral.

[F3]

Integrable real and complex functions, and their integrals: $L^{1}$ and $L^{2}$ denote absolute integrability of the corresponding powers with respect to the indicated measure.

[F4]

Cauchy-Schwarz inequality for $L^{2}$ : the integral Cauchy–Schwarz inequality applies to square-integrable functions.

Proof

Given: [A1] and the process hypotheses in the statement.

1.1A1F1F3F4givenalgebra

By second-order stationarity, $E ∣ V (t) ∣^{2} = \sum_{i} C_{i i} (0) = E ∣ V (0) ∣^{2}$ for every $t$ . Tonelli applied to the nonnegative jointly measurable function $∣ V ∣^{2}$ therefore gives $E \int_{0}^{T} ∣ V (s) ∣^{2} d s = T E ∣ V (0) ∣^{2} < \infty$ . Consequently the path integral of $∣ V ∣^{2}$ is finite almost surely; [F4] on $[0, T]$ then makes $V$ pathwise integrable. Moreover, $∣ X (T) ∣^{2} \leq T \int_{0}^{T} ∣ V (s) ∣^{2} d s$ , so $X (T) \in L^{2}$ .

2.1F1F4step 1.1givenalgebra

The bound in step 1.1 and Cauchy–Schwarz in probability give $E \int_{0}^{T} ∣ V (s) ∣ d s \leq T (E ∣ V (0) ∣^{2})^{1 / 2} < \infty$ . Fubini and $E [V (s)] = 0$ therefore give $E [X (T)] = 0$ . For each $i, j$ , [F4] and stationarity give $E ∣ V_{i} (u) V_{j} (v) ∣ \leq E ∣ V (0) ∣^{2}$ on the finite square $[0, T]^{2}$ . Fubini therefore permits expansion of $E [X_{i} (T) X_{j} (T)]$ as the double integral of $E [V_{i} (u) V_{j} (v)]$ . Splitting the square into $u \leq v$ and $v \leq u$ , then writing $s = ∣ v - u ∣$ , gives the two terms $\int_{0}^{T} (T - s) C_{i j} (s) d s$ and $\int_{0}^{T} (T - s) C_{j i} (s) d s$ . Since the means are zero, this is the stated covariance formula.

3.1A1F2step 2.1

Divide the covariance formula by $2 T$ and extend the integrand by zero for $s > T$ . Its $(i, j)$ entry is $\frac{1}{2} 1_{[0, T]} (s) (1 - s / T) (C_{i j} (s) + C_{j i} (s))$ . As $T \to \infty$ it converges pointwise to $\frac{1}{2} (C_{i j} (s) + C_{j i} (s))$ and is bounded by the integrable function $\frac{1}{2} (∣ C_{i j} (s) ∣ + ∣ C_{j i} (s) ∣)$ . By [F2], each matrix entry converges to the stated integral.

4.1step 3.1algebra∎

For every $a \in R^{d}$ , $a^{⊤} Cov (X (T)) a / (2 T) \geq 0$ because it is a scaled variance. Entrywise convergence from step 3.1 gives $a^{⊤} D a = \lim_{T \to \infty} a^{⊤} Cov (X (T)) a / (2 T) \geq 0$ . Thus $D$ is positive semidefinite. For a scalar velocity observable in units m/s, $D$ has units m $^{2}$ /s.

Remarks

The MIT notes derive a self-diffusion formula from a physical long-time mean-square-displacement assumption. The theorem here proves only the stated covariance limit under second-order stationarity and absolute integrability; it supplies neither that physical identification nor a Gaussian limit.

Green–Kubo covariance limit under absolute integrability

Statement

Facts & Assumptions

Proof

Remarks

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