I use the following definition for a stochastic process.
Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection $X:=(X_t)_{t\in T}$ of random variables is called a stochastic process with state space $(E, \mathcal E)$ indexed by $T$ and defined on $\Omega$ if for all $t \in T$, $X_t: \Omega \to E$ is $\mathcal F/\mathcal E$ measurable. In addition, for $\omega \in \Omega$ the function $t \mapsto X_t(\omega)$ is called the sample path of $X$.
My question is how to interpret the $\omega$ in $X_t(\omega)$ in the above definition. Consider the infinite fair coin tossing experiment. In this case, it is clear that one sample path can be
$$X_t(\omega) = 0101011101010 \cdots, {t \in \{1, 2, 3, \cdots\}}.$$
What exactly is the $\omega$ in this sample path, please? How do I understand it? Thank you!
Best Answer
Let's say we want to model the fivefold throw of a fair coin. Then we define the corresponding process via the outcome of the throws, i.e. we set
$$X_t := \begin{cases} 1 & \text{t-th throw is head} \\ 0 & \text{otherwise} \end{cases}$$
for $t \in \{1,\ldots,5\}$. Now, since we have a fair coin, the probability that $X_t$ equals $1$ is 0.5 for each $t$. In probability theory, this is translated in the following abstract way: For a probability space $(\Omega,\mathcal{A},\mathbb{P})$, the random variables $X_t$ have to satisfy $\mathbb{P}(X_t=1)=\tfrac{1}{2}$. This means in particular that we do not care how the probability space looks like; only the distribution of the random variables is of importance.
Moreover, for any $\omega \in \Omega$ the mapping $t \mapsto X_t(\omega)$ is a realization of our process. If we throw the coin five times and observe e.g. $$0 \, \, 1 \, \, 0 \, \, 0 \, \, 1,$$ then there exists $\omega \in \Omega$ which "symbolizes" this outcome, i.e.
$$(X_1(\omega),X_2(\omega),X_3(\omega),X_4(\omega),X_5(\omega))=(0,1,0,0,1).$$
Usually, we are interested in questions like "What is the probability that we throw head 3 out of 5 times?"; this probability equals $\mathbb{P}(\sum_{t=1}^5 X_t = 3)$. This question can be answered if we know the (finite dimensional) distributions of the stochastic process $(X_t)_t$.
So, basically a stochastic process (on a given probability space) is an abstract way to model actions or events we observe in the real world; for each $\omega \in \Omega$ the mapping $t \mapsto X_t(\omega)$ is a realization we might observe. The likeliness of the realization is characterized by the (finite dimensional) distributions of the process.