68.2. A REMARKABLE THEOREM, HERMITE POLYNOMIALS 2323

Corollary 68.2.4 Let P0n denote all polynomials of the form

p(W (h1) , · · · ,W (hk)) , degree of p≤ n, some h1, · · · ,hk

Also let Pn denote the closure in L2 (Ω,F ,P) of P0n . Then

Pn =⊕ni=0Hi

Proof: It is obvious that Pn ⊇⊕ni=0Hi because the thing on the right is just the closure

of a set of polynomials of degree no more than n, a possibly smaller set than the polyno-mials used to determine P0

n and hence Pn. If Pn is orthogonal to Hm for all m > n, thenfrom the above Theorem 68.2.3, you must have Pn ⊆ ⊕n

i=0Hi. So consider Hm (W (h)) .Recall that Hm is the closure of the span of things like this for |h|H = 1. Thus we need toconsider

E (p(W (h1) , · · · ,W (hk))Hm (W (h))) , |h|H = 1,

and show that this is 0. Now here is the tricky part. Let {e1, · · · ,es,h} be an orthonormalbasis for

span(h1, · · · ,hk,h) .

Then since W is linear, there is a polynomial q of degree no more than n such that

p(W (h1) , · · · ,W (hk)) = q(W (e1) , · · · ,W (es) ,W (h))

Then consider a term of q(W (e1) , · · · ,W (es) ,W (h))Hm (W (h))

aW (e1)r1 · · ·W (es)

rs W (h)r Hm (W (h))

Now from Corollary 64.6.1 these random variables {W (e1) , · · · ,W (es) ,W (h)} are inde-pendent due to the fact that the vector (W (e1) , · · · ,W (es) ,W (h)) is multivariate normallydistributed and the covariance is diagonal. Therefore,

E (aW (e1)r1 · · ·W (es)

rs W (h)r Hm (W (h)))

= aE (W (e1)r1) · · ·E (W (es)

rs)E (W (h)r Hm (W (h)))

Now since r ≤ n, W (h)r = ∑rk=1 ckHk (W (h)) for some choice of scalars ck. By Lemma

68.2.1, this last term,

E (W (h)r Hm (W (h))) = ∑k

ckE (Hk (W (h))Hm (W (h))) = 0

since each k < m.Note how remarkable this is. P0

n includes all polynomials in W (h1) , · · · ,W (hk) someh1, · · · ,hk, of degree no more than n, including those which have mixed terms but a typicalthing in ⊕n

i=0Hi is a sum of Hermite polynomials in W (hk). It is not the case that youwould have terms like W (h1)W (h2) as could happen in the case of Pn.

Obviously it would be a good idea to obtain an orthonormal basis for L2 (Ω,F ,P). Thisis done next. Let Λ be the multiindices, (a1,a2, · · ·) each ak a nonnegative integer. Also inthe description of Λ assume that ak = 0 for all k large enough. For such a multiindex a∈Λ,

a!≡∞

∏i=1

ai!, |a| ≡∑i

ai