Probability 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 generating function (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∼and show that 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 random variable 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 Hilbert space 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 measure for Brownian motion 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 ! 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..