Let hr (x) be the standardized Hermite polynomial of degree r satisfying hr (x)hs (x)(x)dx =

Let hr

(x) be the standardized Hermite polynomial of degree r satisfying ∫

hr

(x)hs

(x)ϕ(x)dx = δrs where ϕ(χ) is the standard normal density. If Xi = hi(Z) where Z is a standard normal variable, show that X1,… are uncorrelated but not independent. Show also that the second cumulant of n 1/2X̅ is exactly one but that the third cumulant does not converge to zero. Construct a similar example in which the first three cumulants converge to those of the standard normal density, but where the central limit theorem does not apply.