Bartlett's Formula for Sample Autocorrelation Variance

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Time Series Analysis.

Your Task
Let $X_1, \dots, X_n$ be a stationary time series with sample autocorrelation function $\hat{\rho}(h) = \frac{\sum_{t=1}^{n-h} (X_t - \bar{X})(X_{t+h} - \bar{X})}{\sum_{t=1}^n (X_t - \bar{X})^2}$.
(1) State Bartlett's formula for the asymptotic covariance $\text{Cov}(\hat{\rho}(h), \hat{\rho}(h+k))$.
(2) Show that for an i.i.d. white noise series, $\sqrt{n} \hat{\rho}(h) \xrightarrow{d} \mathcal{N}(0, 1)$ for any fixed lag $h \ge 1$, justifying the standard $\pm 1.96 / \sqrt{n}$ confidence bands.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.