Intrinsic stationarity of a stochastic processes or random field is a weaker form of stationarity than strict stationarity.

In a strictly stationary process, the entire distribution of the process (i.e., all moments, not just the first two) is invariant under time or space shifts. This means that if you take any collection of points in the process, the joint distributions of these points are the same under any shift in time or space.

On the other hand, an intrinsically stationary (also known as second-order or weakly stationary) process is one where the mean and the variance are constant over time or space, and the covariance between two points depends only on the distance between those points, not on their absolute location. This is a weaker condition because it requires fewer constraints to hold.

Thus, strictly stationary random fields contain the class of intrinsically stationary random fields, not the other way around. Any process that is strictly stationary with a finite second moment is also intrinsically stationary, but the converse is not always true.

$$\text{IntrinsicallyStationary} \subset \text{StrictlyStationary}$$

Strict stationarity (or strong stationarity) is tied to the concept of an invariant distribution over time. In this context, it means that the joint distribution of any set of observations in a stochastic process remains the same even when shifted in time.

To be precise, a stochastic process ${X(t), t ∈ T}$ is strictly stationary if the joint distribution of $(X_{t1}, ..., X_{tn})$ is the same as that of $(X_{t1+h}, ..., X_{tn+h})$ for all choices of time points $t1, ..., tn$ and for all lags $h$. This condition implies that all statistical properties, including mean, variance, higher moments, and covariances, are invariant to time shifts.

For Gaussian processes, this condition is equivalent to second-order (weak) stationarity because a Gaussian process is fully characterized by its first two moments (mean and covariance). Therefore, if these are time-invariant, the whole distribution, hence all moments, are time-invariant.

Stationarity in Stochastic Processes: Second-Order vs Strict Stationarity

In stochastic processes, the idea of stationarity is vital. This property, which ensures that certain statistical properties remain constant over time, simplifies the theoretical analysis of these processes and can often make them easier to model or predict.

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 $t1$ through $tn$ 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