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Stationary Sequence: Definition, Weakly/Strongly

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A real-valued process is a stationary sequence (also called a stationary stochastic process) is defined as one where the following formula is true for every k and m [1]:
stationary sequence

In other words, it is a sequence of random variables that, when shifted, results in the same probability distribution [2]. Informally, you can think of a stationary sequence as modeling successive states of a system in equilibrium [3].

An independent identically distributed (iid) sequence is a special case of a stationary sequence; Stationary is a stronger condition than iid [5].


Strictly and Weakly Stationary Sequence

A sequence is strictly stationary if all n-dimensional distributions of x(t1 + τ),…x(tn + τ) are independent of τ (t is the time lag). A sequence is weakly stationary (or second order stationary) if [4]:


  1. The mean of the sequence is constant, and
  2. It’s covariance is function only of the time lag, t.

Ergodic Stationary Sequence

A stationary sequence is ergodic if it satisfies the strong law of large numbers [6]. With a stationary sequence, we’re only concerned with statistical properties (i.e. that a shifted sequence has the same distribution as the original sequence). With ergodicity, we’re concerned with experimental results, or what you actually observe. A stationary process doesn’t have to be ergodic, but the easiest stationary sequences to work with are, because the theoretical distribution will match your experimental results.

References

[1] Roch, S. (2012). Lecture 13 : Stationary Stochastic Processes. Retrieved April 9, 2021 from: https://www.math.wisc.edu/~roch/teaching_files/275b.1.12w/lect13-web.pdf
[2] Hough, B. (2017). Math 639: Lecture
[3] Fristedt, B. & Gray, L. (1997) Stationary Sequences. In
A Modern Approach to Probability Theory. pp 553-578. Springer.
[4] Lindgren, G. Lectures on Stationary Stochastic Processes. Retrieved April 9, 2021 from: http://www.maths.lth.se/matstat/staff/georg/Publications/lecture2006.pdf
[5] Sigman, K. (2014). 1 Stationary sequences and Birkhoff ’s Ergodic Theorem. Retrieved April 9, 2021 from: http://www.columbia.edu/~ks20/6712-14/6712-14-Notes-Ergodic.pdf
[6] Appendix A Ergodicity, Martingales, Mixing. Retrieved April 9, 2021 from: https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470670057.app1

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Stephanie Glen. "Stationary Sequence: Definition, Weakly/Strongly" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/stationary-sequence/
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