A particularity of random signals, is that the Fourier transform cannot be defined in the same manner, as it will be also a random process (the trajectory beging defined with respect to the frequency).
For stationary random signal, we can define the power spectral density.
Power spectral density Let $X$ a weak sense stationary random signal. Let $$ \hat X_T(\nu) = \int_{0}^{T} X(t)e^{-i2\pi\nu t}\mathrm{d}t $$ if $X$ is a continuous time random signal, or $$ \hat X_T(\nu) = \sum_{n=0}^{T-1} X[n]e^{-i2\pi\nu n} $$ if $X$ is a discrete time random signal. The power spectral density of $X$ is given by $$ S_X(\nu) = \lim_{T\rightarrow +\infty} \frac{1}{T}\text{E}(|\hat X_T(\nu)|^2) $$
In practice, this spectral density can be estimated thanks to the following theorem
Wiener-Khintchine Let $X$ be a (weak sense) stationary signal. Its power spectral density is the Fourier transform of its auto-correlation function. More precisely, for a continuous time random signal $X$ $$ S_X(\nu) = \int_{-\infty}^{+\infty} R_X(t)e^{-i2\pi\nu t}\mathrm{d} t $$ with $S_X(\nu) = \text{E}(|\hat X(\nu)|^2)$ et $R_X(t) = \text{E}(x(\tau)x(t+\tau))$ and $$ R_X(t) = \int_{-\infty}^{+\infty} S_X(\nu)e^{i2\pi\nu t}\mathrm{d} \nu $$ For a discrete time random signal $X$ $$ S_X(\nu) = \sum_{n\in\mathbb{Z}} R_X[n]e^{-i2\pi\nu n} $$ with $S_X(\nu) = \text{E}(|\hat X(\nu)|^2)$ et $R_X[t] = \text{E}(x[\tau]x[t+\tau])$ and $$ R_X[n] = \int_{-\frac{1}{2}}^{\frac{1}{2}} S_X(\nu)e^{i2\pi\nu n}\mathrm{d} \nu $$
Strong white noice Let $X$ be a random signal. $X$ is a white noise in a strong sense if for all $t$ all the $X(t)$ are centered and i.i.d.
Weak white noice Let $X$ be a random signal. $X$ is a white noise in a weak sense if for all $t$ all the $X(t)$ are centered, decorrelated and with a finite variance and i.i.d.
Remark: For Gaussian process, the notions of strong or weak white noise are the equivalent.
Gaussian white noise Let $X$ be a random signak. $X$ is a Gaussian White Noise iff for all $t$, the $X(t)$ are independant and $X(t)\sim\cN(0,\sigma_X^2)$
The autocorrelation function of a Gaussian white noise $X$ is a Dirac: $$ R_X(t) = \sigma_X^2\delta(t) $$ The power spectral density of La densité spectrale de puissance of a gaussian white noise $X$ is a constant $$ S_x(f) = \sigma_X^2 $$