Stationarity

Stationarity in Stochastic Processes: Second-Order vs Strict Stationarity

In stochastic processes, the idea of stationarity is means that certain statistical properties remain constant over time.

However, not all forms of stationarity are the same. We primarily discuss two types: Second-order Stationarity (Weak Stationarity) and Strict Stationarity (Strong or Intrinsic Stationarity).

Second-order Stationarity

A stochastic process ${X(t), t ∈ T}$ is termed second-order stationary or weakly stationary if:

1. The mean function, $E[X(t)]$, is constant.
2. The variance, $Var[X(t)]$, is also constant.
3. The covariance between $X(t)$ and $X(s)$, $Cov[X(t), X(s)]$, depends solely on the difference $(t - s)$, not on the actual time values $t$ and $s$. This implies that the autocorrelation for any pair of time periods remains consistent.

Strict (Intrinsic) Stationarity

A process ${X(t), t ∈ T}$ is strictly stationary (also referred to as strongly stationary or intrinsically stationary) if the joint distribution of any set of observations ${X(t1), ..., X(tn)}$ remains the same as the joint distribution of the shifted set of observations ${X(t1 + h), ..., X(tn + h)}$, for all choices of time points $t_1$ through $t_n$ and all time lags $h$.

This condition requires that all statistical properties of the process, not just the first two moments, are invariant to time shifts, implying that the distribution of the process remains unchanged over time.

Special Case: Gaussian Processes

Gaussian processes present a special case. Such a process is fully described by its first two moments (mean and variance). Consequently, for a Gaussian process, the conditions of second-order stationarity and strict stationarity coincide.

This implies that if a Gaussian process is second-order (weakly) stationary, it is also strictly (strongly) stationary, since once the mean and covariance functions are known and time-invariant, the entire distribution, including all of the moments, is known and time-invariant.

Nevertheless, it is crucial to note that not all second-order stationary processes or strictly stationary processes are Gaussian. The Gaussianity of a process is an additional condition over and above stationarity.

In conclusion, while second-order and strict stationarity are closely connected, they are not identical except in the special case of Gaussian processes.