# Subleading term in the central limit theorem

Let $$X_1,\dots,X_n$$ and $$Y_1,\dots,Y_n$$ be i.i.d. Gaussian random variables of mean zero and variance one.

Then $$Z:= \sum_i \frac{X_i Y_i}{\sqrt{n}}$$ is approximately Gaussian with $$\mathbf E Z =0, \quad \mathbf E Z^2=1.$$ The central limit theorem implies that $$\mathbf E f(Z) =\frac{1}{\sqrt{2\pi}} \int_{\mathbf R} f(x) e^{-x^2/2}\,\mathrm{d}x + o(1)$$ for any smooth bounded $$f$$ as $$n\to\infty$$. Due to the Berry-Essen theorem the convergence rate should be $$n^{-1/2}$$.

Question: Can the subleading term be identified? I.e. can one get $$\mathbf E f(Z) =\frac{1}{\sqrt{2\pi}} \int_{\mathbf R} f(x)e^{-x^2/2}\,\mathrm{d}x + \frac{F(f)}{\sqrt{n}} + o(n^{-1/2})$$ for some $$F(f)$$? If yes, does this $$F(f)$$ depend on the derivative of $$f$$?

• Possibly relevant: the Delta method. – madnessweasley Feb 14 at 23:04
• The integral is missing the Gaussian density inside, no? – leonbloy Feb 15 at 12:48