Stochastic Process vs a sequence of r.v.

Question

A stochastic process $\{X(t) : t \in T \}$ is a collection of random variables. That is for each $t \in T$, $X(t)$ is a random variable. [Sheldon Ross]

I am trying to understand the "collection" part with a sequence of random variables $\{X_n \}$.

I see that $T$ could be uncountable. But why does it make no sense to talk about the convergence of $\{ X_t\}$ ? At first I thought it was because the $X(t)$s have nothing to do with each other, but http://en.wikipedia.org/wiki/Convergence_of_random_variables seems to imply that we don't care if each random variable is different from one another.

EDIT: Recently http://faculty.arts.ubc.ca/vmarmer/econ527/527_07.pdf shows there is such thing. So I am not sure why I was told it was meaningless.

Why do you think that it "make(s) no sense to talk about the convergence of {Xt}"? — Did, Sep 22 '13 at 18:39

score 0 · Accepted Answer · answered Sep 23 '13 at 17:24

0

Each random variable is a mapping from a probability triple $(\Omega,\mathcal{F},\mathbb{P})$ to say $\mathbb{R}$. If you define a stochastic process as just a collection of random variables, then it is not clear whether each of these are defined on the same triple or on different triples. If the triples are different, then it does not make sense to talk about almost-sure convergence, but it is still fine for other types of convergence.

answered Sep 23 '13 at 17:24

galan

174

What other convergence do you have in mind? – Lemon Sep 25 '13 at 07:07
In http://faculty.arts.ubc.ca/vmarmer/econ527/527_07.pdf, there is some meaning for almost-sure convergence. – Lemon Sep 28 '13 at 07:19

Stochastic Process vs a sequence of r.v.

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