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I am having a hard time intuitively understanding the definition of a stochastic process when we fix $\omega$ and let $t$ vary.

More specifically, we know that a stochastic process is a collection of random variables $\{ X(t,\omega)\}$, where $t \in T$ and $\omega \in \Omega$. It’s clear that for every $t$, $X(t, \dot)$ is a random variable.

What I don’t see is how we get a “path” or “trajectory” when we fix $\omega$. For example, suppose we use a simple random walk where we toss a coin, and if the coin lands heads, then $X(heads)=1$ and we move one step up, and, if the coin lands tails, then $X(tails)=-1$ and we move one step down. Our initial point is 0. So, in this case, if I fixed my $\omega$ to be heads, then wouldn’t the trajectory, or path, be represented by the function $f(t)=t$? Alternatively, if we fixed $\omega$ to be tails, then the trajectory would be $f(t)=-t$. Obviously, there are infinite amount of trajectories, or realizations, (going up and down) that repesent this simple walk. So how do I find these deterministic simple paths, by fixing an $\omega$, for this example, when I only have two sample points.

Live Free or π Hard
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  • I don't understand, do you mean you flip one coin and that decides your direction forever, or you flip a coin at each step to decide which way to go? – Ian Nov 13 '17 at 15:12
  • I know my question doesn’t make much sense. What I mean is, what would happen if I looked at $t \mapsto X(., heads)$. I fixed my $\omega$. Now I look at $t=1$, $t=2$ and so on. Then it would be a straight line. – Live Free or π Hard Nov 13 '17 at 15:16

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When you talk about random processes, to fix $\omega$ is not to set "$\omega=\mbox{heads}$" forever. To fix $\omega$ is to set a sequence $\omega=\left(\mbox{heads}, \mbox{tails}, \mbox{tails}, \mbox{heads}, \ldots\right)$.

fonini
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  • So I would have, $X(1,heads), X(2, tails), X(3, tails),...$, discrete time, as my trajectory, or path, when I fixed this $\omega$, which is actually a sequence of outcomes? – Live Free or π Hard Nov 13 '17 at 15:39
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    No, the $\omega$ is that whole sequence, all the random information you need to specify the whole trajectory packed into one object. – Ian Nov 13 '17 at 15:44
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    Beware that you shoudn't write $X(1,heads)$. What's going on is: $$ X(1,(h,t,t,h,\ldots)) = h \ X(2,(h,t,t,h,\ldots)) = t \ X(3,(h,t,t,h,\ldots)) = t \ X(4,(h,t,t,h,\ldots)) = h \ \vdots $$ – fonini Nov 13 '17 at 15:44
  • Perfect! I know we aren’t suppose to say thanks, but this is the best explanation! – Live Free or π Hard Nov 13 '17 at 15:49
  • @fonini If that is a sequence, then $\omega={H, T, H, ...} \notin \Omega={H, T}$? – null Jan 10 '20 at 13:19
  • @NathanExplosion That's exactly the problem. $\Omega$ could not be ${H,T}$. The set $\Omega$ needs to be able to account for all of the randomness in the experiment. In our case, the experiment is to toss the coin an infinite number of times. Therefore, the knowledge of which $\omega\in\Omega$ was the one that happened must be enough to determine the values of all random variables, for instance the values $X\left(1,\omega\right), X\left(2,\omega\right), X\left(3,\omega\right)$ etc, which are the results of the tosses. – fonini Jan 11 '20 at 02:07
  • We might as well make $\Omega$ be the set of all possible sequences of tosses, for example. – fonini Jan 11 '20 at 02:07
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    @fonini two follow-up questions: 1. In the continuous case, would this kind of $\Omega$ be well-defined? 2. If we let $\Omega$ accounts for all possible sequences, then what is the role of sigma-algebra $\mathcal{F}$ (event)? – null Jan 11 '20 at 09:53
  • The question of what should $\Omega$ be when dealing with a continuous-time stochastic process is harder, and the question of what should $\mathcal F$ be is not straightforward even in the present discrete case; I wouldn't be able to provide a satisfying answer in a comment. You should search Math.SE for mentions to "sigma-algebra for continuous stochastic process" or something like that. If you are not satisfied with what you find, you should post it as a question. – fonini Jan 14 '20 at 01:03