I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as:
$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$
for $T^\epsilon < x \ll T^{1/k}$, as $T\rightarrow \infty$
I would appreciate if someone can give me a reference where this calculation is being done (obviously by induction)?
P.S
$\mathbb{E}_{\omega,T}(F(t)):= \int_{\mathbb{R}} \omega \left(\frac{t}{T}\right)F(t) \frac{dt}{T}$
where $\omega$ is a non-negative weight function , whose integral over all the real line is 1, its Fourier transform is well defined, smooth and supported by the interval $[-1,1]$.
Thanks in advance.