Hi guys!!!
A is white noise consisting of independent variables identically distributed Gaussian centered with variance = 1
how to calculate E(A)^{4} ???? E(.) is the Esperance
It's not clear what you mean. By "Esperance" you mean "exected value"? Of course, then "E^4(A)" is not just that number to the fourth power. It is, rather the "4th moment", the integral of x^4 times the probability distribution, $\displaystyle \int_{-\infty}^\infty x^4e^{-x^2}dx$. By symmetry, that is $\displaystyle 2\int_0^\infty x^4e^{-x^2}dx$.
For a standard normal distribution X, we have:
$\displaystyle E(X^4) = {1 \over \sqrt{2\pi}}\int_{-\infty}^\infty x^4 e^{-\frac 1 2 x^2}dx = 3$
as you can see here: Wolfram|Alpha Results
Note that the standard normal distribution has probability density $\displaystyle {1 \over \sqrt{2\pi}}e^{-\frac 1 2 x^2}$.