Well, let me just provide a couple of examples.

For instance, if you consider Yt equal to a zero Xt plus a

one Xt minus one plus unknown plus

an Xt minus n. This is an example of a filter.

And if you take Xt as white noise and a zero equal to one,

so you will get exactly what is called moving average process.

More generally speaking, we have not a moving average,

but the current of this moving average process.

And since this process can be very natural interpreting that you take values of

the process X present and n past time points.

Then you somehow linearly combine these values and again the process way.

Another popular choice is to take Yt as an integral over R exponent into

power minus β (t minus s) Xs ds.

This is a stochastic integral, I discussed previously,

when properties of this object and this is

actually also a transformation of a process X into process Y.

These two examples of filters possess the following properties.

First of all, most of them are linear.

So linearity.

Linear of a filter means that if you take two processes Xt one and Xt two,

and you know that Xt one is transformed into Yt one.

And the process Xt two is transformed into Yt two.

Then any linear combination of Xt one and Xt two

is transformed to linear combination of Yt one and Yt two with the same coefficients.

So, trivially for both of our examples,

this property is fulfilled.

Another property is time invariance.

We say that the process is time invariant if it possesses the following property.

If Xt is transformed to Yt then Xt plus h is transformed to

Yt plus h. And this property should be fulfilled for any

positive h. This property is trivial for our first example,

but for the second it is less trivial.

Let me show argue why this filter is also time invariant.

This is just because,

if you consider integral over R exponent has a power minus B

(t minus s) Xs plus h ds.

This integral is equal to the integral over R exponent minus B(t plus h

minus s plus h) multiplied by X(s plus h) ds.

Then what you have here,

you can change the variables.

So, change s plus h to s,

you will get exactly the process X at the time point t plus h. So,

both of these filters are linear and time invariant.

And then what follows,

let me generalize these types of filters.

In what follows, we'll consider filters in the form

Yt equal to the integral over R. Then we can have

some function rho and the time moment S multiplied by

Xt minus s ds.

Trivially, our second example is exactly of this form.

But as for the first,

we can somehow use indicators to show

that this process is also in this form.

But for simplicity, let me also write a discrete form of this filter.

Actually, it is the same as the sum from h from minus infinity to plus

infinity rho of h Xt minus h. And in this respect,

the function rho is very clearly equal to a one multiplied by

the indicators that h is equal to zero plus

a one multiplied by indicator that h is equal to one and so on.

So, I think that it's very clear that these two cases are analogue of each other.

This is a discrete time filter,

and this is a continuous time filter.

And in what follows,

we'll employ the notion of spectral density for

the analysis of such filters.