In wikipedia page, a random process is a collection of random variables defined on a common probability space $(\Omega,\mathcal{F},P)$ , all take values in the same mathematical space S measurable with respect to $\sigma$-algebra $\Sigma$.

So to my understanding, that means we have a set with element $X:(\Omega,\mathcal{F},P)\rightarrow(S,\Sigma)$. I am quite confident with this part of my understanding. However, I don't quite get what does it mean that the collection of the random variable is indexed by some set $T$, all take values in the same mathematical space $S$.

I always understand the indexation as a kind of 1-1 function $I$, in this example, we have $I: T\rightarrow \mathcal{X}$ where $\mathcal{X}$ is the collection of all $X$

Up to this point, if I were to write the mathematical expression of a stochastic process, I will write: $$\mathcal{X}=\{X_i:i\in T\}$$

However, I am very confused with the difference between a random vector and a stochastic process, in a random vector, we have a series of $X_i,i\in1,2,...,n$ and we will define $X=(X_1,X_2,X_3,...,X_n)$ as a random vector. I know they are definitely not the same thing but I cannot really tell what makes them different. In some textbook, they give notation for stochastic process $X(\omega,t)$ which seems to tell the difference. This means $X$ is not only a function of $\omega$ but also a function of $t$. However, I don't know how to incorporate this way of explanation into my pre-established understanding of a random variable, which as I have explained, is $X:(\Omega,\mathcal{F},P)\rightarrow(S,\Sigma)$.

Sorry if you find my question silly. I have very limited exposure to rigorous mathematics but I'd like to be introduced to all necessary concepts to understand this term. Thank you so much!

  • $\begingroup$ Suppose, a Random Experiment(say tossing a coin) is repeated infinitely many times. And each time/ each trial we are interested in observing the outcome as head(1) and tail(0) . If this outcome is denoted by $X_i$ for the i-th trial then the sequence $X_1,X_2,\dots,$ will be a stochastic process. The beauty of the topic is such that if we now define $$Y_n=f(X_1,\dots,X_n)$$ then $Y_n$ will also form a sequence of random variables which should be Stochastic Process. $\endgroup$
    – FAM
    Apr 28 '21 at 20:45
  • 3
    $\begingroup$ You can definitely view a random vector in $\mathbb{R}^n$ as a stochastic process indexed by $\{1,2,\ldots, n\}$. Vice versa, a stochastic process indexed by $\mathbb{N}$ is an infinite dimensional random vector. $\endgroup$
    – d.k.o.
    Apr 28 '21 at 20:52
  • $\begingroup$ And to add a remark. The index set doesn't need to be countable. In the generic setting of continuous-time stochastic processes your index set will rather be $[0,T]$ or $[0,\infty)$. $\endgroup$
    – Tobsn
    Apr 28 '21 at 20:55
  • $\begingroup$ random vectors and random processes are things completely different, the first is a vector-valued random variable and the second an indexed collection of random variables (vector-valued or whatever) $\endgroup$
    – Masacroso
    Apr 28 '21 at 23:06

In the general case, the index set $T$ needs not be countable. For instance, a stochastic process $X(\omega, t)$ with $t \in T = [0, 1]$ could not be expressed as a random vector.

  • $\begingroup$ realizations of $X(\omega,t)$ could be for example in $L^2([0,1])$ though? $\endgroup$
    – Snoop
    Apr 29 '21 at 11:37
  • $\begingroup$ See here for example. $\endgroup$
    – Snoop
    Apr 29 '21 at 11:50

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