1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

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 $\omega\in\Omega$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.