Stationary time series

Stationary time series:

A time series x(t); t=1,… is called to be stationary if its statistical properties do not depend on time t . A time series may be stationary in respect to one characteristic, e.g. the mean, but not stationary in respect to another, e.g. the variance:

M(x(t)) = const – the mean does not depend on time t;

Var(x(t)) = v(t) – the variance depends on time t;

where M(·) is the mean, Var(·) is the variance.

If joint probability distributions does not depend on time itself but only on the difference of time moments,

 P(x(t)) = p(x);
 P(x(t1), x(t2)) = p(t2 – t1);
 P(x(t1), x(t2), x(t3)) = p(t2 – t1, t3 – t2);
 …

then the time series x(t) is stationary in respect to any statistical characteristic.

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