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A rho-mixing CLT for square-integrable stationary sequences

Statement

Assume AC. Let $(Y_{k})_{k \geq 0}$ be a centered strictly stationary real sequence with $E [Y_{0}^{2}] < \infty$ . If $\sum_{n = 1}^{\infty} ρ_{Y} (n) < \infty,$ then the covariance series $σ_{Y}^{2} : = E [Y_{0}^{2}] + 2 \sum_{k = 1}^{\infty} Cov (Y_{0}, Y_{k})$ converges absolutely and is nonnegative. If $σ_{Y}^{2} > 0$ , then for $S_{n} = \sum_{k = 1}^{n} Y_{k}$ , $n^{- 1 / 2} S_{n} \Rightarrow N (0, σ_{Y}^{2}) .$ If $σ_{Y}^{2} = 0$ , Zero covariance long-run variance gives a degenerate CLT under absolute summability gives the degenerate normal limit.

Facts & Assumptions

Given: AC, the centered strictly stationary real sequence $Y$ , finite second moment, and $\sum_{n \geq 1} ρ_{Y} (n) < \infty$ .

[A1]

AC is the inherited assumption of the local extension and martingale CLT suppliers. (The Axiom of Choice)

[F1]

The coefficient $ρ_{Y} (n)$ bounds absolute correlations between the finite past and separated future. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

The sequence has a two-sided stationary realization $X_{k} = X_{0} \circ T^{k}$ with the same nonnegative-index law and the same maximal-correlation upper bounds for the full past; if $ρ_{Y} (n) \to 0$ , its invertible shift is ergodic even for mod-null invariant events. (Two-sided stationary extension preserves maximal-correlation bounds)

[F3]

Conditional expectation is the $L^{2}$ orthogonal projection onto the conditioning subspace and preserves ordinary expectation. (Conditional expectation is the l2 orthogonal projection, Basic algebra and order properties of conditional expectation)

[F4]

For a centered $L^{2}$ process generated by an invertible probability-preserving ergodic transformation, summability of $∥ E [X_{j} ∣ F_{0}] ∥_{2}$ gives a normal CLT with variance $E [D_{1}^{2}] \geq 0$ , including zero; under absolute covariance summability this variance equals the covariance series. (Summable conditional projections give a stationary martingale CLT)

[F5]

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

Proof

1.1A1F1F2given

Summability of the nonnegative numbers $ρ_{Y} (n)$ implies $ρ_{Y} (n) \to 0$ . By [F2], realize $Y$ as the nonnegative restriction in law of the two-sided stationary sequence $X_{k} = X_{0} \circ T^{k}$ with an invertible probability-preserving shift $T$ that is ergodic for mod-null invariant events. Its full past $F_{k} = σ (X_{i} : i \leq k)$ satisfies $∣ Cov (U, V) ∣ \leq ρ_{Y} (n) \sqrt{Var (U) Var (V)}$ for past $U$ and future $V$ separated by $n$ . All second moments of the coordinates agree with those of $Y$ .

2.1F3step 1.1givenalgebra

Put $W_{j} = E [X_{j} ∣ F_{0}]$ for $j \geq 1$ . By [F3], $W_{j}$ is centered and square-integrable and $E [W_{j} X_{j}] = E [W_{j}^{2}]$ . If $∥ W_{j} ∥_{2} = 0$ , the bound $∥ W_{j} ∥_{2} \leq ρ_{Y} (j) ∥ X_{0} ∥_{2}$ is immediate. Otherwise apply step 1.1's covariance inequality to $W_{j}$ and $X_{j}$ , divide by $∥ W_{j} ∥_{2}$ , and obtain that same bound. Consequently $\sum_{j \geq 1} ∥ E [X_{j} ∣ F_{0}] ∥_{2} \leq ∥ X_{0} ∥_{2} \sum_{j \geq 1} ρ_{Y} (j) < \infty$ .

3.1F4F5step 1.1step 2.1given∎

Put $γ_{j} = Cov (X_{0}, X_{j})$ . Step 1.1 applied to the two coordinates gives $∣ γ_{j} ∣ \leq ρ_{Y} (j) E [X_{0}^{2}]$ for $j \geq 1$ ; if $X_{0}$ is constant, both sides vanish. Thus $\sum_{j \geq 1} ∣ γ_{j} ∣ < \infty$ . All hypotheses of [F4] now hold by steps 1.1 and 2.1. It follows that $n^{- 1 / 2} \sum_{k = 1}^{n} X_{k} \Rightarrow N (0, τ^{2})$ and $τ^{2} = E [X_{0}^{2}] + 2 \sum_{j \geq 1} γ_{j} \geq 0$ . This includes $τ^{2} = 0$ and proves absolute covariance convergence. Since $X$ and $Y$ have the same nonnegative-index law, the series is exactly $σ_{Y}^{2}$ and the finite sums have the original laws. Therefore the asserted nondegenerate and degenerate limits both hold for $Y$ .

Source notes

The proof uses the stronger summable individual-projection criterion, proved locally by an exact martingale-coboundary series. No rho-mixing CLT is imported as a proof premise.

Depends on

Used by

A uniform-mixing CLT for square-integrable stationary sequences Theorem
Sufficient MCMC CLT regimes Theorem

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Sources

Galin L. Jones, On the Markov Chain Central Limit Theorem, Theorem 7 (standard reference, not scraped)
Mikhail Gordin and Magda Peligrad, On the functional central limit theorem via martingale approximation, §§1–2 and §3.3 (standard reference, not scraped)