About non-gaussian noise

unica

Dear friends,I have a question about non-gaussian noise as follow:

R(t) is a white noise and Y1,Y2,Y3 are three uncorrelated Gaussian Variables with average 0 and standard deviation 1, so that
[tex]Z_{1}=\int^h_{0}R(t)dt=h^{1/2}Y_{1}[/tex]
[tex]Z_{2}=\int^h_{0}Z_{1}(t)dt=\int^h_{0}(\int^t_{0}R(s)ds)dt=h^{3/2}(Y_{1}/2+Y_{2}/(2\sqrt{3})[/tex]
I know how to deduce above formulas, as Z1,Z2 are both Gaussian noises, so we can calculate the covariance matrix to get them.

But a question emerged, when I calculated the following formula.

[tex]Z_{3}=\int^h_{0}Z_{1}^2(t)dt=\int^h_{0}(\int^t_{0}R(s)ds\int^t_{0}R(y)d y)dt\approx h^{2}/3(Y_{1}^2+Y_{3}+1/2)[/tex]

Because Z3 is not a gaussian noise, and it refer to calculate the higher moments, so how to deduce the last formula?