Is it possible to have a Random/Stochastic Process composed of different probability density functions?

Question

This question might be a bit too specific but an answer will help me grasp if I understood the concept of random processes correctly. As we may know, a random process $X(t,w)$ where the continuous variable chosen here is time and $w$ is the set of all possible outcomes of my random variable, can be defined from two perspectives:

-As a collection of random variables after fixing the time $t = t_1$ for example, for each outcome $w$

-As a collection of sample functions, where we get continuous functions in t for each outcome $w$

So far my understanding is that, if I take different time instants and compute the ensemble average for each collection of RVs, I would use the same density function for each computation. My question is the following: is it possible to have for a random process at different time instants, assuming the same set of possible outcomes, different probability density functions (i.e at $t= t_1$ it's Gaussian, $t= t_2$ it's uniform, etc.)? Does this imply that my pdf is time varying? In other words, for a "regular" random process, is the PDF always the same but only it's statistics are time varying? Or is there a flaw in my understanding of this concept? I hope this question is clear and please do let me know if this is not the place to post this question, this is the first time on stackExchange!

Answer 1

I'm not entirely sure I understand what you're asking, but there are not really any restrictions on the PDFs of stochastic processes. We can have any distribution for $X_{t_1}$ and any other (or the same) distribution for $X_{t_2}$ (as long as $t_1 \ne t_2$, of course). In general, processes that have the same distribution over time come up less often than processes whose distribution changes over time, i.e. there is no reason to expect that $X_{t_1}$ and $X_{t_2}$ have the same distribution.