We are investigating assumptions on a discrete time process which will permit us to make reasonable estimates of the parameters. We will look for assumptions which guarantee at least the existence
Definition: A stochastic process is stationary if the joint distribution of is the same as the joint distribution of for all t and all k. (Often we call this strictly stationary.)
Definition: A stochastic process
is
weakly (or second order) stationary if
Remark:
Definition: X is Gaussian if, for each the vector has a Multivariate Normal Distribution.
Examples of Stationary Processes:
1) Strong Sense White Noise: A process is strong sense white noise if is iid with mean 0 and finite variance .
2) Weak Sense White Noise:
is second order stationary with
In this course we always use as notation for white noise and as the variance of this white noise. We use subscripts to indicate variances of other things.
Example Graphics:
2) Moving Averages: if is white noise then is stationary. (If you use second order white noise you get second order stationary. If the white noise is iid you get strict stationarity.)
Example proof:
which is constant as required. Moreover:
Most of these covariances are 0. For instance
The proof that X is strictly stationary when the s are iid is in your homework; it is quite different.
Example Graphics:
The trajectory of X can be made quite smooth (compared to that of white noise) by averaging over many s.
3) Autoregressive Processes:
An AR(1) process X is a process satisfying the equations:
Now for
how is Xt determined from
the
? (We want to solve the equations
(1) to get an explicit formula for Xt.)
The case
is notationally simpler. We get
Since
it seems reasonable to suppose that
and
for a stationary series X this is true in the appropriate mathematical sense.
This leads to taking the limit as
to get
Claim: It is a theorem that if is a weakly stationary series then converges (technically it converges in mean square) and is a second order stationary solution to the equation (1). If is a strictly stationary process then under some weak assumptions about how heavy the tails of are converges almost surely and is a strongly stationary solution of (1).
In fact if
are constants such that
and
is weakly stationary (respectively strongly stationary with finite variance)
then
Example Graphics:
Motivation of the jargon ``filter'' comes from physics. Consider an electric circuit with a resistance R in series with a capacitance C. We apply an ``input'' voltage U(t) across the two elements and measure the voltage drop across the capacitor. We will call this voltage drop the ``output'' voltage and denote the output voltage by Xt. The relevant physical rules are these:
These rules give