Understanding stochastic processes.

Question

I'm working through my statistics course which includes some material on stochastic processes. Unfortunately the way it's explained is not giving confidence in my understanding so I would appreciate it if someone could help explain.

From my course notes:

A random process is a time-varying function that assigns to each outcome s a function of time X(t,s) for -T≤t≤T, where 2T is the total observation interval.

• For a fixed sample point s, function X(t,s) versus time is called a sample function of the random process.

• For fixed t: a random process is a random variable.

• If one scans all possible outcomes of the underlying random experiment, we shall get an ensemble of signals.

From this my understanding is that a random process is effectively a random variable that assigns a function of t to each outcome rather than a real number. Is this correct?

Can the functions assigned to each outcome change with time as well? That is, at t=t0 is it possible that there are a different set of functions of t assigned to each outcome than at t=t1? Can the "ensemble of signals" be different at different times? I feel like these may all be the same question phrased differently.

Thanks

Answer 1

Consider a stochastic process like a box of functions of time. The specific kind of boxes you are dealing with contains functions defined over a period 2T. There will be boxes with finite such functions inside, boxes with infinite functions, and one box with every possible function over that period inside.

If you decide to concentrate on a specific function, you are taking an element out of the box and looking at it. This is a sample function (my professor also called it a "realization"). The act of extracting that function out of the box is your random draw.

On the other hand, you can ask yourself: what if I consider all possible values across all functions at a given time $t$? In this case, you potentially get a different value at time $t$ for each function in the box, i.e. you have a random variable. Some functions may have the same value, other might differ. Values might be limited to a few options or vary in a bounded or unbounded range - this depends on the specific box you're looking at.

So, your interpretation is correct: you have a random event s and when you fix it, you get a function out of the box and there's no more randomness.

Your following question is a little inconsistent because you say that you select functions at specific times: each function is defined over the whole range of time.

Nothing prevents you from having lots of functions (or infinite ones) that have the same value at time $t_0$ but have different values at time $t_1$ though, so don't fall into the trap of thinking that knowing a value at $t_0$ lets you know all the rest. Depending on the process, this might be true or not, so it's not true in the general case.