I am learning from Knill's probability theory and stochastic processes with applications book. In chapter 4 about "continuous time stochastic processes" I encountered the following.

For any fixed $\omega \in \Omega$, one can regard $X_t(\omega)$ as a function of $t$. It is called a sample function of the stochastic process. In the case of a vector-valued process, it is a sample path, a curve in $ \mathbb{R}^d$

However, I am a bit confused. Do they mean that a sample function assigns to every value $t$ a unique random variable $X : \omega \rightarrow \mathbb{R}$ ? Or what does the sample function actually do?

I tried to find more information about what a sample function is, and came on a math encyclopedia website, though the information that is given there still confuses me. After reading that information I still wonder: which values is this function mapping to which co-domain.

  • $\begingroup$ If parameter $t$ takes values in set $T$ and $X_t:\Omega\rightarrow\mathbb R^d$ for each $t\in T$, then any fixed $\omega\in\Omega$ induces a function $T\rightarrow\mathbb R^d$ prescribed by $t\mapsto X_t(\omega)$. $\endgroup$ – drhab Mar 15 '15 at 13:31

The idea is that you fix some $\omega\in\Omega$. Then you may consider the map $X(\omega)\colon \mathbb{R}^+\rightarrow \mathbb{R}$ given by $X(\omega)(t) = X_t(\omega)$. For each $\omega$ this gives you some real function, where such a function is called a "sample path" or " sample function"

Edit: To elaborate, the sample function thus gives you the development of the random variables $X_t$, for some chosen $\omega$. It is possible to plot the sample function, but there is such a function for each $\omega\in\Omega$.

  • $\begingroup$ That edit also made things very clear to me. So you can see how a probability distribution of a given random variable changes over time. Thanks. $\endgroup$ – Pedro Mar 15 '15 at 13:12

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