I would like to evaluate
$$ F = \frac{\mathbb{E} \left\{\left(\int_0^T x^3(t) dt \right)^2\right\}}{\mathbb{E} \left\{\left(\int_0^T x(t) dt \right)^2 \right\} } \approx \frac{\mathbb{E} \left\{\left(\frac{T}{N} \sum_{n=1}^N x_n^3 \right)^2\right\}}{\mathbb{E} \left\{\left(\frac{T}{N}\sum_{n=1}^N x_n \right)^2 \right\} } $$
(The second term with the sum is an acceptable approximation if the continuous way causes problems)
If $x(t) \sim \mathcal{N}(0, \sigma^2)$ and independent for each $t$, the denominator is a random walk and the output variance is proportional to the integration length $T$, i.e., the denominator would be something like $\sim \sigma^2 T$.
However, $x^3(t)$ is not Gaussian so I cannot evaluate the numerator. Any clues?
Generally, I would like to extend the evaluation to cases where $x(t)$ is not independent for each $t$ (i.e., colored noise or bandlimited white Gaussian noise).
The context is as follows: I have a nonlinear amplifier (for simplicity only $y(t)= a_0 x(t) + a_3 x^3(t)$ followed by an ideal integrator. I would like to calculate the expected mean squared error after $T$ seconds if the input signal follows certain statistics, in the simplest case it is white Gaussian.
The normalized mean squared error in dB can be obtained as
$$ \mathrm{NMSE} = 20\log(4/3) - 2 \, \mathrm{IIP_3} + 20 + F $$
Let me also add that in the absence of the integral, simply $F = \frac{\mathbb{E} \left\{\left(x^3(t)\right)^2\right\}}{\mathbb{E} \left\{\left(x(t)\right)^2 \right\} } = 15 \sigma^4$, although the The Cube of a Normal Distribution is Indeterminate.