I'm reading the book Probability, Random Variable & Stochastic Processes by Papoulis & Pillai 4th Ed.

Chapter 9 starts with definition of Stochastic Process. On Pg 373 Sec 9-1 Definitions, it states

As we recall, a random variable x is a rule for assigning to every outcome $\zeta$ of an experiment S a number $\bf{x}(\zeta)$. A stochastic process $\bf{x}$$(t)$ is a rule for assigning to every $\zeta$ a function $\bf{x}$$(t, \zeta)$.

Then it goes on to add the following:

Thus a stochastic process is a family of time functions depending on the parameter $\zeta$ or, equivalently, a function of t and {. The domain of $\zeta$ is the set of all experimental outcomes and the domain of t is a set $\mathbb{R}$ of real numbers.

Thereafter, there is a series of contradictions that I feel with this definition. The definition means that for if conditionally say $\zeta_i$ occurs then the function $\bf{x}$$(t)$ results which on this given condition is now completely deterministic, rather than a collection of random variables defined on the same sample space. In some places in the book this the latter view is implicit. For example on pg. 375 titled $\bf{Statistics\ of\ Stochastic\ Processes}$

A stochastic process is a noncountable infinity of random variables, one for each $t$.

Towards the end of pg 373,

We shall use the notation $\bf{x}$$(t)$ to represent a stochastic process omitting, as in the case of random variables, its dependence on $\zeta$. Thus $\bf{x}$$(t)$ has the following interpretations:

  1. It is a family (or an ensemble) of functions x(t, $\zeta$). In this interpretation, t and $\zeta$ are variables.
  2. It is a single time function (or a sample of the given process). In this case, t is a variable and $\zeta$ is fixed.
  3. If t is fixed and $\zeta$ is variable, then x(t) is a random variable equal to the state of the given process at time t.
  4. If t and $\zeta$ are fixed, then $\bf{x}$$(t)$ is a number

For point 2, this means that a sample path $x_i(t)$ when $\zeta = \zeta_i$
is completely deterministic & not a collection of random variables. Say $\zeta \in \{H,T\}$ be the sample space. Let the stochastic process be, $x(t,H) = 1+t, \ t\geq 0$ & $x(t,T) = 1-t, \ t\geq 0$. Thus $P\{x(t) = 1+t |\ \zeta=H\} = 1$, is completely deterministic.

Thus, this definition only limits itself to these type of stochastic processes.

This has left me confused.


1 Answer 1


I think the author simply explains a convenient abuse of notation.

The stochastic process is a map $(t,\zeta)\mapsto x(t,\zeta)$ (or equivalently, a family $(t\mapsto x(t,\zeta))_{\zeta}$).

  1. Have you ever written something like "the function $t^2$ is differentiable" ? Well, $t^2$ is technically not a function, but $t\mapsto t^2$ is, and that is what you meant, and everybody understood it. Similarly, you might write $x(t,\zeta)$ to denote the function $(t,\zeta)\mapsto x(t,\zeta)$. However in many reasonings, the variable $\zeta$ is often fixed and it is annoying to write $\zeta$ everywhere. So we agree to say that $x(t)$ denotes $x(t,\zeta)$, which itself might denote the function $(t,\zeta)\mapsto x(t,\zeta)$.

  2. If $\zeta$ is fixed, you might want to look at the function $t\mapsto x(t,\zeta)$ (usually called a sample path). Recall that we are use to writing $x(t)$ instead of $x(t,\zeta)$. So you want to have a look at the function $t\mapsto x(t)$ (the variable $\zeta$ is implicit). For convenience, we denote by $x(t)$ the map $t\mapsto x(t)$.

  3. You also might want to have a look at the map $\zeta\mapsto x(t,\zeta)$, where $t$ is fixed. Even if $\zeta$ is not fixed this time, we are really used to making it implicit and we still denote by $x(t)$ the map $\zeta\mapsto x(t,\zeta)$, which is a random variable.

  4. Of course, if $t$ and $\zeta$ are fixed, then $x(t,\zeta)$ is a number, which is denoted $x(t)$ where we omit $\zeta$, as usual.

In the end, depending on the context, $x(t)$ might denote a stochastic process, a sample path, a random variable or a number. Usually, the context is clear enough so that we do not get confused. If the context becomes ambiguous, then you should specify your notation (for instance write $t\mapsto x(t,\zeta)$ instead of $x(t)$, since the latter might be misunderstood).

  • $\begingroup$ I have edited my question for clarity. My doubt is that the definition is very restrictive. $\endgroup$
    – Ricky
    Mar 16, 2019 at 13:34
  • $\begingroup$ "the definition is very restrictive". Which definition? Point 2? It's not really a definition. It's more like one of the 4 possible ways of understanding the term $x(t)$. Depending on the context, $x(t)$ means different things. Each situation described by points 1 to 4 is different. Of course in the case where $\zeta$ is fixed, everything is deterministic (sort of), but that concerns only points 2 and 4. $\endgroup$
    – Will
    Mar 16, 2019 at 14:22

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