Hello, I am having trouble understanding the nomenclature associate with a stochastic process, especially when associated with a time series.

Let or simply be a stochastic process. Then, we can describe the process by finding the joint pdf of observations taken at times , for any value of . So, my stochastic process looks like ?

I guess I am not understanding the relationship between and the subscript k as in, the difference between seeing and .

My thought is,

we have a stochastic process , which is a colection of observations taken times

x--------------------x--------------------x----------------------x

where the x denotes observations (or realizations) , , , and respectively through some time . So, for instance, if I am measuring human traffic in airports and I collect my sample data at each one of these times and come up with some kind of distribution for each sample (X_{t_{1}},...,X_{t_k}) then try to find a joint distribution to describe this particular process, which I let run for a given amount or time. Is this correct?

So, a process, such as a random walk

, where Z_{t} is white noise is made of two process; namely and , correct? Do I veiw as a random variable with a joint distribution made up of random variables observed times for a duration, so that, is a process?

As you can tell, my thoughts are pretty scattered. I would really appreciate some help understanding the basic structure of stochastic process relating to time series analysis.

Thank you