Random process and time series

Question

How to understand what is in front of me - a time series or a random process? For example, if I measure the temperature outside every day, is this a random process or a time series?

Answer 1

If you take time-ordered measurements (such as the daily temperature), you end up with a finite sequence of numbers $T:= (t_1,...,t_n)$. Such a sequence of empirical measurements is referred to as time series.

If you want to model the process which produced T, you can do so by using mathematical constructs such as stochastic processes or dynamical systems.

Whether the time series T is, lets say, a stochastic process is more of a (somewhat ill-defined) philosophical question. Rather you should ask, whether the real-world process which produced T can be modelled as a stochastic process (or dynamical system, etc) in a useful manner. So it boils down to not confuse real-world measurements with mathematical modelling tools.

Which tool you should use to model $T$ depends on a million things (last but not least your research goal) and unfortunately cannot be broken down into a simple set of rules.