68.2. A REMARKABLE THEOREM, HERMITE POLYNOMIALS 2321

Consider m < n

∂ n+m

∂ ns∂ mtexp(stE (XY )) =

∂ m

∂ tm ((tE (XY ))n exp(stE (XY )))

You have something like

∂ m

∂ tm [tn ((E (XY ))n exp(stE (XY )))]

and m < n so when you take partial derivatives with respect to t, m times and set s, t = 0,you must have 0. Hence, if n > m,

E (n!Hn (X)m!Hm (Y )) = 0

Similarly this equals 0 if m > n. So assume m = n. Then you will go through the sameprocess just described but this time at the end you will have something of the form

n!E (XY )n + terms multiplied by s or t

Hence, in this case,E (n!Hn (X)n!Hn (Y )) = n!E (XY )n

and soE (Hn (X)Hn (Y )) =

1n!

E (XY )n

Let W be the function defined above, W (h) is normally distributed with mean 0 andvariance |h|2 and E (W (h)W (g)) = (h,g)H . Then from Lemma 68.2.1,

E (Hn (W (h))Hm (W (g))) =

{0 if n ̸= m

1n! (E (W (h)W (g)))n if n = m

=

{0 if n ̸= m

1n! (h,g)

nH if n = m

This is a really neat result. From definition of W,

E((W (h)W (g))1

)= (h,g)H

Note this is a special case of the above result because H1 (x) = x. However, we don’t knowthat E ((W (h)W (g))n) is equal to something times (h,g)n

H but we know that this is true ofsome nth degree polynomials in W (h) and W (g).

Definition 68.2.2 Let Hn ≡ span{Hn (W (h)) : h ∈ H, |h|H = 1}.

Thus Hn is a closed subspace of L2 (Ω,F ). Recall F ≡ σ (W (h) : h ∈ H). This sub-space Hn is called the Wiener chaos of order n.

Theorem 68.2.3 L2 (Ω,F ,P) =⊕∞n=0Hn. The symbol denotes the infinite orthogonal sum

of the closed subspaces Hn. That is, if f ∈ L2 (Ω) , there exists fn ∈Hn and constants suchthat f = ∑n cn fn and if f ∈Hn,g ∈Hn, then ( f ,g)L2(Ω) = 0.