I am a little confused about random vectors and stochastic processes. I read their definitions in Wikipedia (random vector,stochastic process) and I cannot understand their differences . I would appreciate your help. Thanks.
A random vector is a generalization of a single random variables to many. A stochastic process is a sequence of random variables, or a sequence of random vectors (and then you have a vector-stochastic process).
When we use the concept of a stochastic process to model events that evolve through time, then there is no way you should be able to confuse them: Taking a certain sub-sequence of the stochastic process does not form a "random vector" -because with time as the framework, for a collection of random variables to form a random vector, they must all "happen" in the same time period (they must have the same time index). Index-wise, the specific position of each random variable in a random vector is arbitrary - but position is critically defining in a time-stochastic process.
Things may get a bit blurry when the stochastic process is used to model random variables that do not have different time indices (e.g. cross-sectional r.v.'s). It is not common to think of such collections of r.v.'s as "stochastic processes", but it is perfectly legitimate. Here, the index may not be critical in the case of the SP also, so one may think "what is the difference if I call this collection a stochastic process or a random vector?" Well, the basic difference is that a stochastic process has an open dimension - you can think of "adding" rv's to the sequence, and you will still have the "same" SP, just with more realizations. A "random vector" has a fixed dimension, and if you add one r.v. to it, you get a different random vector.
$\begingroup$ Thanks a lot. just a question are the random variables in a stochastic process identically distributed? how about random vector? $\endgroup$– AlexNov 18, 2013 at 16:43
$\begingroup$ Not necessarily and not necessarily. Usually of course we tend to consider stochastic processes and random vectors that contain r.v's with identical distributions (or identical up to some important moments, eg. in weakly stationary processes, distributions have the same first and second moments). $\endgroup$ Nov 18, 2013 at 16:58
$\begingroup$ I read their definition in Wikipedia again. According to Wikipedia, random variables are on the same probability space, does this mean that they are identically distributed? another question is that are these random variables dependent? $\endgroup$– AlexNov 19, 2013 at 2:39
1$\begingroup$ The "same probability measure" refers to the joint distribution function - not to the marginal distribution each r.v. may have. These marginals can be anything, and a set of marginals can give rise to any number of joint probability distributions. $\endgroup$ Nov 19, 2013 at 4:08
1$\begingroup$ Defining a random process as a collection of random vectors what stop us from "unpacking" the vectors to a collection of random variables? $\endgroup$ Jul 19, 2022 at 18:50