Can a random variable be viewed as a stochastic process where at any moment in time the distribution is independent of others and the same as others?

Question

• I.e. can the random variable be viewed as a more specific instance of a stochastic process -- a sort of constant stochastic process?
• I know that for analysis textbooks often look at a stochastic process as a collection of random variables at different moment in time (I think this makes analysis easier). But let's say the time steps becomes infinitely small and we have infinitely many distributions -- to me it looks like a single random variable, whose domain is simply a function of time. So by that logic if at any moment in time the distribution is the same, then the domain is the same and it looks like a simple random variable where the domain is constant (not a function of time).

Answer 1

To answer your first question, you certainly can view it that way, but I'm not sure what you get out of it.

To answer your second question (which may help with your first question), a random variable is usually defined as a mapping from a probability space to some measurable space of values. If you make the domain a (non-constant) function of time, then it seems to me that makes it by definition not a single random variable. A stochastic process adds the element of time, as well as introducing such properties as the history of the process, on which the current value can depend in any of a range of ways.

If it is a constant function of time, then I suppose you can treat it as samples of a single random variable. It would help if I understood better what you think can be gained by looking at it this way.