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Home , Hermite polynomials

Limit Theorems, II, Homework 4, Hermite polynomials and Wiener chaos

Probabilists often use a different normalization for Hermite polynomials than the one in homework 3. Adopting this, we define Hn(x)by

x2/2 x2/2 Hn(x)e− = ∂xe− . (1)

n n It is common to put in a factor of ( 1) so that Hn has the form x + .Our H is x3 +3x,notx3 3x. − ··· 3 − − 1. The exponential (as opposed to the “ordinary” generat- ing function) for the Hn is

∞ zn F (x, z)= H (x) . n! n nX=0 The ordinary generating function is missing the factor 1/n!. Use the for- mula (1) to find F (x, z) explicitly. Use this to find an integral formula for Hn(x). This formula can be used to derive approximations to Hn(x) when n and x are large.

2 2. With the notation g(x)= 1 e x /2, show that √2π −

Hn(x)Hm(x)g(x)dx =0 whenm = n, Z 6 and evaluate H (x)2g(x)dx . Z n

k x2/2 x2/2 Hint: Show that x e− Hk(∂x)e− (e.g. using the Fourier trans- form). Suppose that m

n x2/2 n m x2/2 Hm(x)∂x e− = ∂x − P (∂x)e− ,

where P is some polynomial. For m = n this is almost true.

3. Show that the Hermite polynomials are a (complete) basis for the Hilbert 2 space Lg, which has inner product

u, v = u(x)v(x)g(x)dx . h ig Z

Do do this, compute the Hermite polynomial expansion of eiξx (formula (1) will be helpful here) and show explicitly that it converges to eiξx.Why is this enough? 4. Suppose that X is a standard normal and f(x)has E(f 2(X)) < . Interpret the Hermite expansion of f as a representation of the random∞ variable f(X) as a linear combination of “Hermite” random variables Hn(X). 5. Here is the extension to n variables. A multi-index is a list of n non negative α α1 α integers: α =(αn, ,αn). The notation x means x x n ,and ··· 1 ····· n ∂α = ∂α1 ∂αn . The degree of a multi index is p = α = α + + α . x x1 xn 1 n The multidimensional····· Hermite polynomials are | | ···

x 2/2 α x 2/2 Hα(x)=Hα (x ) Hα (xn)=e| | ∂ e−| | . (2) 1 1 ····· n x 2 n Show that the Lg(R ) is a direct orthogonal sum of the degree p subspaces

p =spanof Hα α = p . S { ||| } 2 n 6. For any orthogonal n n matrix, Q, and any function f Lg(R ), define Qf by (Qf)(x)=f(×Qx). Clearly Qf = f . ∈ k kg k kg a. Verify by direct calculation that the space is invariant under the S2 action of Q for n = 2. That is, show that if x0 = x cos(θ)+y sin(θ) and y0 = y cos(θ) x sin(θ), then − H2(x0)=a(θ)H2(x)+b(θ)H1(x)H1(y)+c(θ)H2(y) ,

and find a similar representation for H1(x0)H1(y0).

b. Give a simpler proof that each p is invariant for any n.Hint:what α S does Q do to ∂x ? iξ x c. Compute fp(x, ξ)= pe · . Here, I have used to denote the orthog- onal projection ontoS the space . Hint: theS right Q can make this easy. S 7. Now let (Ω, ,µ) represent Wiener for on the F interval [0, 1] with W (0) = 0. Let L be the algebra of sets generated by the diadic interval differences F ⊂F

L L L GL,k = W ((k +1)2− ) W (k2− ) , for k =0, 1,...,2 1. − − For any f(W ) (I will use W instead of ω to represent the basic random element of Ω: an element of Ω is a path.) with E[f 2(W )] < ] define ∞ fL,p(W )= pE[f L] . S |F Here, we use p, as in part 6c, to be the orthogonal projection onto the space of orderSp in the space of functions of n =2L independent gaussians. Show that, for each p, fL,p is a martingale as a function of L. Show that the limit lim fL,p(W )=fp(W ) L →∞ exists, that f(W )= p∞=0 fp(W ), and that P

2 ∞ 2 E f (W ) = E fp (W ) .   Xp=0  

8. Show that for each f and p there is a function ap(t1,...,tp)with

1 t1 tp 1 − fp(W )= ap(t1,...,tp)dW (tp) dW (t1) . Z0 Z0 ···Z0 ···

Hint: Take a limit of approximations aL,p(t1,...,tp), which are martin- gales (in L)foreachp as functions of (t1,...,tp) (show this). Note that 2 in the limit, the energy in all terms involving H2(x)=x 1 or higher goes to zero. These spaces are the Wiener chaos spaces of degree− p.