Ergodic / stationary

[Vidhya Gholkar]

Reply to the question by stee...@conrad.appstate.edu

No, Ergodic and stationary processes are distinctly entities.

Suppose you want to find the mean of some continious random process. Then by
definition you integrate it's pdf over all values :
+oo
E(x) = int f(x)dx (1)
-oo

The time time average of a single realisation is given by

+T
avg(x(t)) = lim (1/2T) int x(t) dt (2)
t -> oo -T

So if (1) approaches (2) with probability 1 we have a mean ergodic process.

If all the statistical moments are the same as the time averaged
moments we have an ergodic process. So an ergodic process is one
whose time averages equal its ensemble average. This basically means
you can swap a time average for an ensemble average ... which means
you do not have to have access to all possible realisations otherwise
we would be in big trouble most of the time ...

A stationary process is one whose statistical properties are
invariant to a shift in origin.

Often people call a process whose mean is constant and whose
autocorrelation is a function of the lag between two points
stationary; strictly speaking this is incorrect and infact they are
referring to a wide sense stationary process.

A stationary process must be widesense stationary but not vice versa.

An ergodic process must be stationary but not vice-versa. To make
life simple people invariably assume that the stationary process they
have is ergodic so then from a single realisation of the process they
can make estimates of the statistical property that interests them.

Ergodicity is more complicated really !!!!! A sufficient condition for a
process to be ergodic is that a random process be regular and
stationary and well enough behaved that you can swap the
expectation and time average operators. What's regular ? time averages
of any realisation are the same ! What's well behaved ? That's where it
gets tougher .... (Birkoff in 1931 wrote a paper on this published in
Proc. National Academy of Science ....)

I think Doob in his classic text "Stochastic Processes" J. Wiley
treats it rigoursly .... but, I I don't have access to one at the
moment.

Vidhya Gholkar

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[Vidhya Gholkar]

Oops, of course, my message should say if (2) approaches (1) .....