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Zero covariance long-run variance gives a degenerate CLT under absolute summability

Statement

Let $(Y_{k})_{k \geq 0}$ be a centered, strictly stationary, square-integrable real process (The space $L^{p} (μ)$ as the quotient by null functions), and put $γ_{k} = Cov (Y_{0}, Y_{k})$ . If $\sum_{k \geq 1} ∣ γ_{k} ∣ < \infty$ and $γ_{0} + 2 \sum_{k = 1}^{\infty} γ_{k} = 0,$ then $n^{- 1 / 2} \sum_{k = 1}^{n} Y_{k} \to 0$ in $L^{2}$ and hence in distribution to $N (0, 0)$ .

Facts & Assumptions

Given: A centered strictly stationary real process with finite second moment and the stated covariance hypotheses.

[F1]

Variance is the centered second moment and covariance is the centered product expectation. (Moments, variance, and covariance on a probability space)

[F2]

Strict stationarity makes every covariance depend only on the lag. (Strong, uniform, and maximal-correlation mixing coefficients)

[F3]

Convergence in $L^{2}$ implies convergence in probability. ( $L^{p}$ convergence implies convergence in probability)

[F4]

Convergence in probability to a constant implies convergence in distribution to its point-mass law. (Convergence in probability implies convergence in distribution)

[F5]

Dominated convergence applies to a series indexed by counting measure when its terms have a summable majorant. (Dominated convergence)

Proof

1.1F1F2F5given

Expanding the finite square of $S_{n} : = \sum_{k = 1}^{n} Y_{k}$ and using centering and [F2] gives $\frac{1}{n} Var (S_{n}) = γ_{0} + 2 \sum_{k = 1}^{n - 1} (1 - \frac{k}{n}) γ_{k} .$ For each fixed $k$ , the weight tends to $1$ and its absolute value is at most $1$ . The summable sequence $∣ γ_{k} ∣$ dominates the summands, so [F5] gives $n^{- 1} Var (S_{n}) \to γ_{0} + 2 \sum_{k \geq 1} γ_{k} = 0$ .

2.1F1F3F4step 1.1∎

Since $E [S_{n}] = 0$ , step 1.1 gives $E [(S_{n} / \sqrt{n})^{2}] = n^{- 1} Var (S_{n}) \to 0$ . This is convergence to zero in $L^{2}$ ; [F3] gives convergence in probability, and [F4] gives convergence in distribution to the point mass at zero, namely $N (0, 0)$ .

Depends on

Used by

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Sources

Galin L. Jones, On the Markov Chain Central Limit Theorem, §4.1 (standard reference, not scraped)
Charles J. Geyer, Markov Chain Monte Carlo Lecture Notes, §4.8 (standard reference, not scraped)