Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

According to the definition of stochastic process in Wikipedia:

Given a probability space $(\Omega, \mathcal{F}, P)$, a stochastic process (or random process) with state space $X$ is a collection of $X$-valued random variables indexed by a set $T$ ("time"). That is, a stochastic process $F$ is a collection

$\{ F_t : t \in T \}$

where each $F_t$ is an $X$-valued random variable.

Does that mean for all values of the index $t$, the state space $X$ and the $\sigma$-algebra on it must be same? For a stochastic process, Can it be allowed that the state space and the $\sigma$-algebra on it is different for different value of the index?

Is it correct that, as a stochastic process, "a random walk on a graph" seems to have different state spaces at different stages/times?

Thanks and regards!

share|improve this question

2 Answers 2

up vote 3 down vote accepted

Yes, the measurable space $(X, \mathcal{M})$ ($M \subset 2^X$ being a $\sigma$-algebra) should be the same for all $t$ in the index set. If you want the process to occupy different sets $X_t$ at different times, then the right things to do is to take $X$ to be the disjoint union of all the $X_t$. There is no requirement that $F_t$ be surjective, so there is no harm if some of the state space is not used at some times.

For a random walk on a graph, the state space $X$ is just the vertex set of the graph, typically with the discrete $\sigma$-algebra. Then for each $t \in T$, $F_t$ is some random element of $X$, i.e. a random vertex of the graph. Formally, if your underlying probability space is $(\Omega, \mathcal{F}, P)$, then $F$ is a map from $\Omega \times T$ into $X$, with the property that for each $t \in T$, the map $\Omega \ni \omega \mapsto F(\omega, t) \in X$, denoted $F_t$, is measurable.

share|improve this answer

It's strange that it's mentioned the same space. In fact random process is a collection $(X_t|t\in \mathcal{T})$ where $X_t\in \mathbb{X}_t$ and state spaces can be different. As I remember sometimes notion "stochastic process" for the cases when all state spaces are the same. You can find a lot of literature on the random process with changing state spaces.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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