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Stationarity

Module by: Nick Kingsbury. E-mail the author

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Summary: This module introduces stationarity, such as strict sense stationarity (SSS) and wide sense stationarity (WSS).

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Stationarity in a Random Process implies that its statistical characteristics do not change with time. Put another way, if one were to observe a stationary random process at some time tt it would be impossible to distinguish the statistical characteristics at that time from those at some other time t′ t′ .

Strict Sense Stationarity (SSS)

Choose a Random Vector of length NN from a Random Process:

X=Xt1Xt2…XtNT X X t1 X t2 … X tN (1)
Its NNth order cdf is
F X ( t1 ) ,   …   X ( tN ) x1…xN=PrXt1≤x1…XtN≤xN F X ( t1 ) ,   …   X ( tN ) x1 … xN X t1 x1 … X tN xN (2)
Xt X t is defined to be Strict Sense Stationary iff:
F X ( t1 ) ,   …   X ( tN ) x1…xN=F X ( t1 + c ) ,   …   X ( tN + c ) x1…xN F X ( t1 ) ,   …   X ( tN ) x1 … xN F X ( t1 + c ) ,   …   X ( tN + c ) x1 … xN (3)
for all time shifts cc, all finite NN and all sets of time points t1…tN t1 … tN .

Wide Sense (Weak) Stationarity (WSS)

If we are only interested in the properties of moments up to 2nd order (mean, autocorrelation, covariance, ...), which is the case for many practical applications, a weaker form of stationarity can be useful:

Xt X t is defined to be Wide Sense Stationary (or Weakly Stationary) iff:

  1. The mean value is independent of tt, for all tt
    EXt=μ X t μ (4)
  2. Autocorrelation depends only upon τ=t2−t1 τ t2 t1 , for all t1 t1
    EXt1Xt2=EXt1Xt1+τ= r X X τ X t1 X t2 X t1 X t1 τ r X X τ (5)
Note that, since 2nd-order moments are defined in terms of 2nd-order probability distributions, strict sense stationary processes are always wide-sense stationary, but not necessarily vice versa.

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