The file covers the main features of a stationary time series.
An accompanying Rmd file is provided.
There are three conditions for a time series to be stationary.
All subpopulations of $X_t$ have the same mean for each time $t$ i.e. the mean does not depend on time.
$$ E[X_t] = \mu $$Variance of $X_t$ does not depend on time and the variance is finite.
$$ Var[X_t] = \sigma^2 < \infty $$The correlation of $X_2$ and $X_1$ only depends on $t_2-t_1$. That is, the correlation between the data points depends only on how far a part the obervations are in time, not where the observations are located in time.
$$ Corr(X_t, X_{t+h}) = \rho_h $$where $h$ is the difference between two points in time.
For constant mean and variance, plot the time series and visually assess the validity of the assumption. For the constant autocorrelation, split the time series into subpopulations and plots the ACFs for the subpopulations, then visually assess the validity of the assumption.
A single realization can be used to estimate the mean, variance, and autocorrelation of a stationary time series when the autocorrelation approach zero as the lag increases.
Parameters can be estimated in the following order due to dependence:
Let $X_t$ be a discrete stationary time series for the estimate procedures.
Then $\bar{x}$ is an unbiased estimate of the mean of the time series. Thus, the mean will be estimated as
$$ \hat{Mean}(X_t) = \bar{x} = \frac{\sum^N x_i}{N} $$Recall the definition of $\rho_k$:
$$ \rho_k = \frac{\gamma_k}{\gamma_0} = \frac{E[(X_t-\mu)(X_{t+k}-\mu)]}{\sigma_x^2} $$The autocorrelations can be estimated with the sample autocorrelations:
$$ \hat{\rho}_k = \frac{\hat{\gamma}_k}{\hat{\gamma}_0} = \frac {\frac{1}{N} \sum^{N-k} \left( x_t - \bar{x} \right) \left( x_{t+k} - \bar{x} \right) } { \frac{1}{N} \sum^{N} \left( x_t - \bar{x} \right)^2 } $$Note that the amount of data used to estimate the sample autocorrelations decreases as $k \rightarrow N$.
The variance of the can be estimated by
$$ \hat{Var}(X_t) = \frac{\hat{\sigma}^2}{N} \sum^{N-1}_{k=-(N-1)} \left(1-\frac{\lvert k \rvert}{N} \right) \hat{\rho}_k $$simplifying the summation by noting that $\rho_k = \rho_{-k}$,
$$ = \frac{\hat{\sigma}^2}{N} \left( 1 + 2 \sum^{N-1}_{k=1} \left(1-\frac{\lvert k \rvert}{N} \right) \hat{\rho}_k \right) $$where $\hat{\rho}_k$ represents the $k$th autocorrelation estimate and $\sigma^2$ is estimated with all the data as shown below.
$$ \hat{\sigma}^2 = \frac{1}{N} \sum^N \left( x - \bar{x} \right)^2 $$The confidence interval for the mean follows the common form of
$$ CI: (Mean \, Estimate) \pm (Multiplier) (Variance \, Estimate) $$Substituting for the estimation, we have
$$ CI: \bar{x} \pm t_{1-\frac{\alpha}{2}} \sqrt { \frac{\hat{\sigma}^2}{N} \left( 1 + 2 \sum^{N-1}_{k=1} \left(1-\frac{\lvert k \rvert}{N} \right) \hat{\rho}_k \right) } $$This confidence interval is interpreted as "We are $X\%$ confident that the mean is contained in the interval $[CI \, Lower, \, CI \, Upper]$"
[1] W. Woodward and B. Salder, "Stationary", SMU, 2019