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


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...?

  • $\begingroup$ See 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. $\endgroup$ – whuber Nov 6 '20 at 12:38
  • $\begingroup$ The link didn't help me much. Thanks anyway. $\endgroup$ – Miss Q Nov 6 '20 at 14:54
  • 1
    $\begingroup$ 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. $\endgroup$ – whuber Nov 6 '20 at 14:58
  • $\begingroup$ @whuber, edited, can you understand now? $\endgroup$ – Miss Q Nov 6 '20 at 15:08
  • $\begingroup$ @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 ). $\endgroup$ – mlofton Nov 6 '20 at 16:39

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.

  • 1
    $\begingroup$ 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. $\endgroup$ – whuber Nov 6 '20 at 18:20
  • $\begingroup$ Hello, it does not fully answer my question, Thanks. $\endgroup$ – Miss Q Nov 7 '20 at 11:25
  • $\begingroup$ Can you write your last statement in mathematical terms, based on my scheme? Thanks. $\endgroup$ – Miss Q Nov 7 '20 at 11:39
  • $\begingroup$ Can you write a matrix form based on my experiment hence a sample path and ....? $\endgroup$ – Miss Q Nov 7 '20 at 11:39
  • $\begingroup$ 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. $\endgroup$ – abstrusiosity Nov 7 '20 at 14:01

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