Is white noise WSS by nature or not?
White noise is not &quot;WSS by nature&quot; whatever you mean by that phrase but it can be treated as a (zero-mean) WSS process insofar as its effects in linear systems are concerned.
For example, standard linear system theory ways when the input to an LTI system is an ordinary WSS process $\{X(t)\}$ with autocorrelation function $R_X(\tau)$, then the output of the LTI system is a WSS process
$\{Y(t)\}$ with autocorrelation function $R_Y(\tau)$ given by
$$R_Y = h\star \tilde{h} \star R_X \tag{1}$$
where $h(t)$ is the impulse response of the LTI system and $\tilde{h}(t) = h(-t)$ is the time-reversed impulse response of the LTI system. The power spectral densities are related as
$$S_Y(f) = |H(f)|^2S_X(f)\tag{2}$$
where $H(f)$ is the transfer function of the LTI system. If $\{X(t)\}$ is a white noise process with autocorrelation function $K\delta(\tau)$ and we pretend that $(1)$ and $(2)$ are still applicable, we get that $\{Y(t)\}$ is a zero-mean WSS process with autocorrelation function $R_Y = K\cdot h\star \tilde{h}$ and power spectral density $S_Y(f) = K\cdot |H(f)|^2$. Of course, mathematicians would laugh at this calculation but physical experiments using the naturally occurring thermal noise in electrical circuits as a stand-in for a white noise process show that these results are pretty close to reality. As engineers, we seek equations that match the universe as we observe it (physicists seek universes that match their equations while mathematicians don't care) and so we go blithely on our way treating white noise as a WSS process in linear systems and everything works out OK. The troubles start when we start treating white noise as a WSS process in nonlinear systems and the world comes crashing down about our ears and we need to start paying attention to what the math people are saying.
White noise is referred as white Gaussian noise if we pretend or claim or insist that $\{Y(t)\}$ is a Gaussian process which means that not only are all the random variables $Y(t)$ Gaussian random variables, but every finite set $\{Y(t_1), Y(t_2), \cdots, Y(t_n)\}, n \geq 2,$ of random variables is a set of jointly Gaussian random variables. Standard random process theory says that when a Gaussian process is passed through an LTI system, the output is a Gaussian process but this fact does not allow us to reverse-engineer the result and claim that all the $X(t)$'s are also Gaussian random variables.