Re: the connection orthogonal expansions and stochastic processes

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 10/18/04 

Date: Mon, 18 Oct 2004 15:37:50 +0100

Wilbert Dijkhof wrote:

>
>> You want all moments to be finite, of course. As long as the distribution
>> is not concentrated on a finite set, there will be such a sequence.
>
> Is it always possible to construct such a sequence? Perhaps you have
> a reference if this construction is not trivial?

It's fairly trivial. The measure induces an inner product
on the polynomial ring: <f,g> = E(f(X)g(X)). This is positive
semidefinite and only fails to be positive definite
if E(f(X)^2) = 0 for some nonzero f, which happens iff the probability
measure is concentrated on the zeroes of f. Apart from
this exceptional case the orthogonal polynomials can be
constructed by the Gram-Schmidt process.

For a very brief treatment of orthogonal polynomials but with
lots of references, see pp. 20-21 of Krattenthaler's
_Advanced Determinant Calculus_
http://www.mat.univie.ac.at/~slc/wpapers/s42kratt.html 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_

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