Thread: Need help with a time series question

Hi Guys,

I cannot get past this question. For the first part I have been calculating the variance of X(t) so as to get it in the form similar to the result the question looks for but I can't quite get it out.

Any help would be much appreciated

Nappy

Hey Nappy.

Assuming I.I.D for the e(t) processes, you can use Var[aX + bY] = a^2Var[X] + b^2Var[Y] where a and b are non-deterministic quantities not related to the X's or Y's in any way (completely independent).

Since you know the sequence is summable, then it should be part of l^2 which means that the inner product of <x,x> (x is your sequence represented as an infinite vector) should also be finite (summable) so the sum of variances should be summable.

Is it enough to say as we know the sequence is summable <x,x> will also be summable and therefore will be less than infinity?

It should be, but I don't know what your lecturer expects.

You may have to use properties that relate l^1 norms to l^2 norms. l^1 is just that the normal series is summable and l^2 is the space where the squared terms are summable. There are theorems in mathmatics that relate the various norms and l^1 norm is finite, then l^2 should be as well.

Since I don't know what your lecturer expects, you will have to think about what is adequate and what isn't.