We distinguish two types of stationarity: strict stationarity (or strong stationarity) and weak stationarity.
Strictly Stationary Process (strongly stationary process) is a stochastic process whose finite-dimensional distributions do not change with time. Formally speaking, a stochastic process X(t) is strictly stationary if, for any moments of time t1, ..., tk, positive shift h and values x1, ..., xk,
P(X(t1+h) ≤ x1, ... , X(tk+h) ≤ xk) = P(X(t1) ≤ x1, ... , X(tk) ≤ xk).
Stationary Process (weakly stationary process) is a stochastic process whose expectations and covariances do not change with time. Formally speaking, a stochastic process X(t) is weakly stationary if, for any moments of time s < t and positive shift h,
1] E[X(t+h)] = E[X(t)],
2] Cov[X(s+h), X(t+h)] = Cov[X(s+h), X(t+h)].
Strict stationarity implies weak stationarity.
STATIONARY PROCESS REFERENCES
Lawler, G. F. (1995). Introduction to Stochastic Processes. New York: Chapman and Hall/CRC.
Ross, S. M. (1995). Stochastic Processes (2nd ed). New York: Wiley.
Karlin, S., & Taylor, H. M. (1975). A First Course in Stochastic Processes (2nd ed). New York: Academic Press.
Gikhman, I. I., & Skorokhod, A. V. (2004). The Theory of Stochastic Processes I. Springer-Verlag Berlin Heidelberg.
Gikhman, I. I., & Skorokhod, A. V. (2004). The Theory of Stochastic Processes II. Springer-Verlag Berlin Heidelberg.
Gikhman, I. I., & Skorokhod, A. V. (2007). The Theory of Stochastic Processes III. Springer-Verlag Berlin Heidelberg.
Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press.
Brockwell, P.J., & Davis, R. A. (1991). Time Series: Theory and Methods (2nd ed). New York: Springer.
Tsay, R. S. (2005). Analysis of Financial Time Series. New Jersey: Wiley-Interscience.
Greene, W. H. (2011). Econometric Analysis (7th ed). Upper Saddle River, NJ: Prentice Hall.
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