The Wikipedia page on stochastic processes defines a stochastic process as:

A stochastic process is defined as a collection of random variables defined on a common probability space (Ω,F,P), where Ω is a sample space, F is a σ-algebra, and P is a probability measure; and the random variables, indexed by some set T, all take values in the same mathematical space S, which must be measurable with respect to some σ-algebra Σ.

Now, unfortunately, this definition confuses me more than it enlightens me. I am confused by the difference between Ω and S, and F and Σ. It seems to me that the definition of a sample space is the set of all of possible outcomes/values, so the definitions of Ω and S identical to me.

Thanks for your effort! I have already searched this website to see if the question already exists, but I could not find it.

  • $\begingroup$ Suppose $(\Omega,\mathscr{F})$ and $(T,\mathcal{T})$ are measurable spaces. A stochastic process with values in $T$ and indexed by $I$ is a function $X:\Omega\rightarrow T^I$ that is measurable. The measurable structure on $T^I$ is that of the product $\sigma$-algebra, that is the $\sigma$-algebra on $T^I$ generated by the sets of the form $A_{i_1}\times \ldots\times A_{i_n}$, where $A_{i_\ell}\in\mathcal{T}$. $\endgroup$ May 6 '21 at 7:01

Let's consider tossing a coin. $X_t$ will denote the t-th toss. For the sake of simplicity I will only consider 2 coin tosses ($t \le 2$)

  • We have elementary events: $\omega$ for example: $(H, H), (H, T), ...$

  • Then the sample space is the set of all $\omega$ (\Omega). We also have events ($A$) which are asubset of $\Omega$.

  • $F$ is a $\sigma$-algebra which contains measureable events. For example one possible choice of events is $F=\{\emptyset, \Omega\}$ where we can only measure these 2 trivial events.

  • $P$ is the probability measure, which is a function $\Omega \to [0,1]$, it tells you the probability of each $\omega$

Up until now we did not talk about $S$ or $\Sigma$

  • $S$ is some valued space, usually $R \text{, or } R^n$
  • A random variable is measureable function $\Omega \to S$.
  • Random variables generates a sigma algebra, which is the most simple sigma algebra for which it is measurable to. For example $X_1$ is measurable on the $\sigma$-algebra $\sigma(X_1)=\big\{ \{(H,H), (H,T)\}; \{(T,H), (T, T)\}, \emptyset, \Omega \big\}$

I hope this helps clarify things. I'd recommend to look into meassure theory and probability theroy before jumping to stochastic processes.


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