根据定义，彩色高斯噪声是广义平稳(WSS) 过程；即具有恒定均值（构成过程的所有随机变量均值相同），其自相关函数仅取决于差的论点。通常使用来表示差异，并通过编写而不是更冗长的来滥用符号来表示自相关函数. 该过程的功率谱密度 (PSD) 就是的傅里叶变换： $R_X(t_1, t_2) = E[X(t_1)X(t_2)]$ $t_2-t_1$$\tau$$t_2-t_1$$R_X(\tau)$$R_X(0, \tau) = R_X(t,t+\tau)$$R_X(\tau)$

$$S_X(f) = \int_{-\infty}^\infty R_X(\tau)e^{-j2\pi ft} \,\mathrm dt.$$ The PSD is an even nonnegative function of $f$.

White noise is a zero-mean process for which $R_X(\tau) = K\delta(\tau)$ where $\delta(\cdot)$ is the Dirac delta or impulse and its PSD has constant value $K$ for $-\infty < f < \infty$. Colored noise is a zero-mean process whose PSD is not constant for all $f$. Colored Gaussian noise is a process in which all the random variables are zero-mean correlated (jointly) Gaussian random variables with random variables separated by time $\tau$ having covariance $R_X(\tau)$. Note that the variance of all the random variables is $\sigma^2 = R_X(0)$. The PSD has the connection to the PDF that the PSD determines the variance of the random variables in question via the following corollary to the inverse Fourier transform formula:

$$\sigma^2 = R_X(0) = \int_{-\infty}^\infty S_X(f) \,\mathrm df.$$ Note that all the random variables constituting the process have the same (Gaussian) PDF (and so the same mean and same variance) and the variance is not a time-varying function due to the noise being colored.

Assuming $E\{ x\}=0$

$$E\{|x|^2\}=r_{xx}(0)=\int_{-\infty}^{\infty} S(f) df$$

See

https://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem

The PDF is Gaussian, what other PDF are you asking about?

For example, consider a discrete-time integrator

$$X_{k+1} = X_k + U_k$$

where $X_0 =: x_0$ is the (zero-variance) initial condition and $U_k \sim \mathcal N (0, \sigma^2)$ is the AWGN input.

$$\begin{array}{rl} X_1 &= x_0 + U_0\\ X_2 &= x_0 + U_0 + U_1\\ X_3 &= x_0 + U_0 + U_1 + U_2\\ &\vdots\\ X_n &= x_0 + U_0 + U_1 + U_2 + \cdots + U_{n-1}\end{array}$$

Hence,

$$\mathbb E (X_n) = x_0 \qquad \qquad \qquad \mbox{Var} (X_n) = n \sigma^2$$

Since the addition of independent Gaussian random variables is still Gaussian, we conclude that

$$X_n \sim \mathcal N (x_0, n\sigma^2)$$

Linear systems preserve "Gaussian-ness". Sometimes, one can do without the frequency domain.