Stationariness

Stationariness is a property of certain stochastic processes, which are mathematical objects used to model random phenomena that evolve over time. A stochastic process is considered stationary if its statistical properties, such as its mean, variance, and autocorrelation, do not change over time. This means that the way the process behaves randomly is consistent from one point in time to another.

There are different types of stationariness. Strict stationariness requires that the joint probability distribution of the process at any set of time points is the same as the joint probability distribution at any other set of time points shifted by the same amount. This is a very strong condition. A weaker, but more commonly used, condition is weak stationariness (also known as covariance stationariness). For a process to be weakly stationary, its mean must be constant, its variance must be finite and constant, and the autocovariance between any two time points must depend only on the time difference between them, not on their absolute positions in time.

Stationary processes are important in many fields, including time series analysis, signal processing, and econometrics. Their statistical properties being time-invariant simplifies analysis and allows for forecasting and modeling. For example, if a time series is stationary, past patterns can be used to predict future behavior. Many real-world phenomena are not perfectly stationary, but they can often be transformed or modeled using techniques that account for non-stationarity, such as differencing or modeling the process as a combination of stationary and non-stationary components.