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When the long-run variance of a stationary process is positive, matching the variance of its average to an average of independent observations gives $T_{\mathrm{eff}}\simeq T\gamma (0)/{\sum }_h\gamma (h)$. Positive aggregate correlation reduces this size; negative aggregate correlation can increase it beyond the observation count. This extends the variance interpretation of effective sample size of a Markov chain.

For coefficients ${\theta }_0=1,{\theta }_1,\dots ,{\theta }_q$, the long-run variance of a stationary process sampled every second time is ${\sigma }^2[({\sum }_{j\text{ even}}{\theta }_j{)}^2+({\sum }_{j\text{ odd}}{\theta }_j{)}^2]$. It equals ${\sigma }^2[\Theta (1{)}^2+\Theta (-1{)}^2]/2$.

For a causal autoregressive model of order $p$, the population partial autocorrelation function cuts off after lag $p$, whereas the autocorrelation function generally decays. For an invertible moving-average model of order $q$, the autocorrelation function cuts off after lag $q$, whereas the partial correlation coefficients generally decay. For a mixed autoregressive moving-average model, both generally decay. Exponential decay may alternate in sign or show damped oscillations. Sample correlation coefficients only approximate these patterns, so isolated crossings of the significance bands are not exact order tests.

In PDF Figure 1, the autocorrelation function alternates sign, with a large negative lag-one correlation coefficient and geometrically decreasing magnitude. The partial autocorrelation function has essentially one substantial spike, negative at lag one and about $-0.8$; later values mostly lie within the displayed bands. Thus a stationary AR(1) with a negative coefficient is the natural first model, with $\hat{\varphi }$ initially near $-0.8$. The trace fluctuates around an approximately constant mean and shows no evident deterministic trend.

Fit that model, compare nearby low-order alternatives using likelihood function and an Akaike information criterion or Bayesian information criterion, and inspect the residual autocorrelation function and residual variance. A few small later PACF spikes are expected in a plot with many lags. The roughly $\pm 1.96/\sqrt{T}$ white noise bands are a guide, rather than simultaneous guarantees for every lag.

The sign alternation also suggests concentration of spectral power near the high-frequency end: the AR(1) denominator $1+{\varphi }^2-2\varphi \cos \lambda$ is smallest near $\lambda =\pi$ when $\varphi <0$. From estimated $\gamma (0)$ and lag-one correlation coefficient one can estimate innovation variance as ${\hat{\sigma }}_{\epsilon }^2\approx \hat{\gamma }(0)(1-{\hat{\varphi }}^2)$. The persistence parameter controls the decay rate and long-run variance of a stationary process. These plots do not establish a normal distribution, independence, or the absence of nonlinear dependence; correlation coefficients of squared residuals can provide a separate variance diagnostic.

First, the absolute values in the printed limit are an error: the correct long-run variance of a stationary process is the signed sum of autocovariances. Directly,soEach coefficient tends to $1$ and has absolute value at most $1$. Absolute summability and the dominated convergence theorem therefore giveThe sum of absolute values is an upper bound, not the general limit. For a concrete counterexample to the printed assertion, take $X_t={\eta }_t-{\eta }_{t-1}$ with unit-variance iid noise. Then $\gamma (0)=2$, $\gamma (\pm 1)=-1$, and other covariances vanish. The absolute sum is $4$, but ${\overline{X}}_T=({\eta }_T-{\eta }_0)/T$, so $T\mathrm{Var}({\overline{X}}_T)=2/T\to 0$.

For correlated data the central limit theorem can still hold under suitable strong mixing of a stationary process and moment conditions, but the limiting variance is the long-run variance of a stationary process, rather than the one-observation variance. When it is positive,Positive serial dependence usually increases the standard error, while negative dependence can reduce it. The approximate effective sample size of a stationary sample is $T{\gamma }_X(0)/{\sum }_h{\gamma }_X(h)$ when that denominator is positive. A long-memory time series can require a different normalization or a different limit law. Absolute covariance summability alone does not prove a CLT: if $X_t=V{\eta }_t$ with a common independent random scale $V$ taking values $1$ and $2$ with equal probabilities and iid ${\eta }_t$ with the standard normal distribution, the off-diagonal covariances are zero, yet $\sqrt{T}{\overline{X}}_T$ has the nonnormal scale-mixture law $VN(0,1)$. The example is not an ergodic stationary process. If the long-run variance of a stationary process is zero, the usual nondegenerate square-root-$T$ CLT is unavailable.

For the causal AR(1), $|\varphi |<1$ andThe conditional distribution follows because the current innovation is independent of the past.

For a fixed known $\varphi$, condition on the observed $X_1$ and use the $n=T-1$ transitions $t=2,\dots ,T$. Their conditional maximum likelihood criterion is, up to constants,Differentiating in $\mu$ yieldsSince $X_t-\varphi X_{t-1}=(1-\varphi )\mu +{\epsilon }_t$,Thus it is an unbiased estimator, even conditionally on $X_1$, andIt has statistical consistency with mean-square convergence, and the iid-noise strong law of large numbers also gives almost-sure statistical consistency. Its conditional distribution is exactly normal with the displayed mean and variance. If $X_0$ is also observed and all $T$ transitions are used, replace $T-1$ by $T$.

The fixed-$\varphi$ qualification is necessary for the exact finite-sample claims. If $\varphi$ is jointly estimated, conditional likelihood function is linear regression with an intercept: writing ${\bar{X}}_{-}=n^{-1}{\sum }_{t=2}^T{X}_{t-1}$ and ${\bar{X}}_{+}=n^{-1}{\sum }_{t=2}^T{X}_t$, the unconstrained estimators areThis ratio is not generally an unbiased estimator and does not have the preceding finite-sample variance. Under the usual stationary regression conditions it has statistical consistency; its asymptotic variance is ${\sigma }_{\epsilon }^2/(1-\varphi {)}^2$. Indeed, it differs from ${\bar{X}}_{+}$ by $\hat{\varphi }(X_T-X_1)/(n(1-\hat{\varphi }))$, an asymptotically negligible endpoint term. Profiling an unknown innovation variance does not change the fixed-$\varphi$ estimate of $\mu$.

Let $Y_i$ be iid with density $f$, set $H_i=h(Y_i)$, and put $I=E_{f}h(Y)$ and ${\tau }^2={\mathrm{Var}}_f(h(Y))$. The finite-second-moment assumption gives, for ${\tau }^2>0$,The sample variance $s_N^2=(N-1{)}^{-1}{\sum }_i(H_i-{\hat{I}}_N{)}^2$ consistently estimates ${\tau }^2$. The central limit theorem and Slutsky theorem therefore give the asymptotic confidence intervalIts confidence probability is approximately $1-\alpha$, conventionally written $100(1-\alpha )\%$; the printed $(1-\alpha )\%$ omits the factor $100$. For ${\tau }^2=0$, $h(Y)=I$ almost surely and the estimator has zero error. This interval concerns iid Monte Carlo estimators; correlated simulation output requires a long-run variance of a stationary process estimate instead of the iid sample variance.

In a second-order sense a weakly stationary process has short memory when its autocovariance is absolutely summable. Then its spectral density of a stationary process is continuous and its sample-mean variance has the usual finite long-run variance of a stationary process. A central limit theorem still needs additional conditions.