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I am trying to understand the definition of a stochastic process and related terminology by myself. I found this intuitive:

http://www.eco.uc3m.es/~jgonzalo/teaching/PhDTimeSeries/StochasticProcessesExamples.pdf

Let $\Omega=\left\{\omega_{1}, \omega_{2}, \ldots\right\}$ and let the time index $0\le n\le N<\infty$, A stochastic process in this setting is a two-dimensional array or matrix such that:

$X=\left[\begin{array}{ccc} X_{1}\left(\omega_{1}\right) & X_{1}\left(\omega_{2}\right) & \ldots \\ X_{2}\left(\omega_{1}\right) & X_{2}\left(\omega_{2}\right) & \ldots \\ \ldots \ldots & \ldots & \ldots \\ X_{N}\left(\omega_{1}\right) & X_{N}\left(\omega_{2}\right) & \ldots \end{array}\right] $

In the context of this (example 1 in the linked pdf), suppose my $\Omega=\{1,2,3,4, 5\}=\{\omega_1,\dots,\omega_5\}$ and time index set $\mathbb T=\{t_0=0,1,2,3,\dots, \infty\}$

and my scheme is I start with a fixed deterministic point $x$ at time $t=t_0$ say, from $\mathbb R$, and then I chose $\omega_1$ at random and define $X_1= f_{\omega_1}(x)$, then next $X_2= f_{\omega_2}(X_1)$ and so on, i.e $X_{n}=f_{\omega_n}(X_{n-1})$, where $\omega_1,\dots$ are i.i.d discrete random variables taking values in $\{1,2,\dots, 5\}$ and I have given $f_1,\dots, f_5$ real-valued functions beforehand. Now in this context, could anyone tell me what is my sample path and what is my random variable and how would I construct a matrix representation like example 1 in the note?

$\star$ $\star$ Also, a little doubt: In the representation of $X$ in matrix form in example1, is the $\omega_1$ in the first entry $X_1(\omega_1)$ in the first row is same as the first entry $X_2(\omega_1)$ in the 2nd row? Or do we just denote by $\omega_1$ which comes as a result of the random experiment no matter whether we are doing the experiment first time or 2nd time or so on? Also, I am thinking that we get the first row when we do the random experiment the first time and 2nd row when we perform the random experiment 2nd time...?

Alexis
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Miss Q
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  • See https://stats.stackexchange.com/questions/126791 for a relevant definition. As far as your matrix representation questions go, please explain in your post what you mean by that, because any external link is liable to disappear at any time. – whuber Nov 06 '20 at 12:38
  • The link didn't help me much. Thanks anyway. – Miss Q Nov 06 '20 at 14:54
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    I'm glad you looked, but telling us it didn't help you won't help us figure out how to answer your question. It looks like you are focusing on issues of some kind of a "matrix representation," but it's essential that you explain or define what that means right here within your post. – whuber Nov 06 '20 at 14:58
  • @whuber, edited, can you understand now? – Miss Q Nov 06 '20 at 15:08
  • @Miss Q: It's an interesting document and I've never seen a stochastic process defined that way before. I only read example 1 carefully because I don't have time right now. But. it explains that each column of the matrix represents a trajectory (i.e: a sample path ). So, one element, $\omega_{j}$ maps to one whole trajectory. So, you don't want to make the new observation a function of the previous observation because the value of $\omega_{j}$ already tells what you the values of $X_{i} ~ i = 1,\ldots N$ are for that path (i.e.; the column of the matrix ). – mlofton Nov 06 '20 at 16:39

1 Answers1

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The pdf you linked says,

Each row represents a random variable and each column is a sample path or realization of the stochastic process.

I would take that to mean that each $\omega_i$ determines a complete sample path $X_1,X_2,\ldots,X_N$.

The formulation given is more abstract than the usual presentation of Markov Processes. We don't have any functional relationship to take us from $X_i$ to $X_{i+1}$. The matrix simply represents the (abstract) dependency between $X$s by their shared dependency on a single $\omega$.

I'm gonna add this for emphasis: The sample path is the result of one experiment (i.e., one $\omega$), not a sequence of experiments.

abstrusiosity
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    One reason this presentation may be more abstract is that Markov processes are an extremely special example of stochastic processes. Many of the simplifications available for Markov processes don't apply generally. There's a deeper problem here: the matrix explicitly supposes there are at most a countable number of sample paths. That's rarely the case. – whuber Nov 06 '20 at 18:20
  • Hello, it does not fully answer my question, Thanks. – Miss Q Nov 07 '20 at 11:25
  • Can you write your last statement in mathematical terms, based on my scheme? Thanks. – Miss Q Nov 07 '20 at 11:39
  • Can you write a matrix form based on my experiment hence a sample path and ....? – Miss Q Nov 07 '20 at 11:39
  • It would be missing the point to apply your scheme to this example. Your scheme is a first order Markov Process. The example is not. It's more general. It's conceptual rather than mechanical. It's a simple starting point for talking about general stochastic processes, not a exercise to be solved. – abstrusiosity Nov 07 '20 at 14:01
  • Okay, then can you tell me what is a sample path, and what is a random variable for my scheme? If possible a little elaborately? Thanks. – Miss Q Nov 09 '20 at 14:40
  • Your scheme is incorrect because it refers to different $\omega$s in the same sample path. To use your scheme you would need each $\omega$ to be a vector of length $n$, so $\Omega$ would have to be $\{1,2,3,4,5\}^n$. – abstrusiosity Nov 09 '20 at 15:07
  • why length $n$? should not $\omega=(\omega_1,\omega_2,\dots)$ be in $\{1,\dots,5\}^{\infty}$? – Miss Q Nov 09 '20 at 15:35
  • Yeah, that right. It should be $\infty$, not $n$. – abstrusiosity Nov 09 '20 at 15:39
  • Or more precisely $\Omega= \{1,\dots,5\}^{\infty}\times \mathbb R$? – Miss Q Nov 09 '20 at 15:39
  • what is an experiment in my case then? – Miss Q Nov 09 '20 at 15:40
  • $\mathbb R$ is the sample space, not the probability space. In your setup $\Omega = \{1,\ldots,5\}^\infty$. An experiment is one realization of the stochastic process. One $\omega$ which produces the sequence $X_1(\omega),X_2(\omega),\ldots$, where each $X_i : \omega \mapsto \mathbb R$. – abstrusiosity Nov 09 '20 at 15:49
  • Could you please tell me what is the object $X_1(\omega)$? Is your $\omega$ an infinite tuple vector? – Miss Q Nov 09 '20 at 15:52
  • $X_1(\omega)$ is the random variable that is the first in the infinite sequence of a sample path and $\omega$ is, as discussed above, an infinite sequence of integers in $\{1,\ldots,5\}$. – abstrusiosity Nov 09 '20 at 15:55
  • Okay so if $\omega=(\omega_1,\omega_2,\dots, \omega_n, \dots)$ Then a sample path is also an infinite sequence $(X_1(\omega_1), X_2(\omega_2), \dots, X_n(\omega_n), \dots)$ right? – Miss Q Nov 09 '20 at 16:14
  • To be consistent with the notation in the initial example it would be better to add another index to $\omega$. One experiment is a draw of one $\omega_i \in \Omega$ which produces, according to your scheme, the sample path $(X_1(\omega_{i1}),X_2(\omega_{i2}),\ldots,X_n(\omega_{in}),\ldots)$. – abstrusiosity Nov 09 '20 at 17:03