Understanding Stationary Time Series: Key Definitions and Yule‑Walker Equations

Stationary Time Series

Here, stationarity refers to wide‑sense stationarity, whose property is that the statistical characteristics of the series do not change with a shift in time, i.e., the mean and covariance remain constant over time.

Basic Concepts and Theory

Definition 1: Given a stochastic process {X_t}, let μ(t) be its mean function and σ²(t) its variance function; these are called the mean and variance functions of the process.

The square root of the variance function is called the standard deviation function, representing the deviation of the process from its mean function.

Definition 2: For a stochastic process {X_t}, fixing a lag τ, define its autocovariance function γ(τ) = Cov(X_t, X_{t+τ}).

This function characterizes the correlation between times t and t+τ, and can be standardized to define the autocorrelation function.

Thus, the autocorrelation function is the standardized autocovariance function.

Definition 3: Let a random sequence {X_t} satisfy (1) a constant mean and (2) be uncorrelated across time; then {X_t} is called a stationary random sequence (stationary time series), abbreviated as a stationary sequence.

Definition 4: For a stationary sequence, its autocovariance function is denoted γ(·).

If the autocovariance function takes the form γ(τ)=0 for τ≠0, the sequence is called a stationary white‑noise sequence. Its variance is constant, and any two distinct time points are uncorrelated. Stationary white noise is the most basic stationary sequence.

Definition 5: Let ε_t be zero‑mean stationary white noise. If a sequence {X_t} satisfies … (specific conditions omitted), then …

Define a random linear sequence. Under condition (18.13) one can prove that the term in (18.14) is a stationary sequence. If a zero‑mean stationary sequence can be expressed in the form of (18.14), this form is called the transfer function, also known as the Green function.

Definition 6: Let {X_t} be a zero‑mean stationary sequence. From the viewpoint of time‑series forecasting, the partial autocorrelation function is defined. Given known values of … we aim to forecast … This leads to a linear minimum‑mean‑square estimator, choosing coefficients …

Expanding the expressions yields the Yule‑Walker equations, which characterize the partial autocorrelation function of the stationary sequence.