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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 $bfx(zeta)$. A stochastic process $bfx$$(t)$ is a rule for assigning to every $zeta$ a function $bfx$$(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 $mathbbR$ 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 $bfx$$(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 $bfStatistics of Stochastic Processes$

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 $bfx$$(t)$ to represent a stochastic
process omitting, as in the case of random variables, its dependence
on $zeta$. Thus $bfx$$(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 $bfx$$(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, $x(t,H) = 1+t, tgeq 0$ & $x(t,T) = 1-t, tgeq 0$. Thus $Px(t) = 1+t = 1$, is completely deterministic.

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

This has left me confused.

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 $(tmapsto x(t,zeta))_zeta$).

1. Have you ever written something like "the function $t^2$ is differentiable" ? Well, $t^2$ is technically not a function, but $tmapsto t^2$ 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 $tmapsto x(t,zeta)$ (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 $tmapsto x(t)$ (the variable $zeta$ is implicit). For convenience, we denote by $x(t)$ the map $tmapsto x(t)$.




3. You also might want to have a look at the map $zetamapsto x(t,zeta)$, 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 $zetamapsto x(t,zeta)$, 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 $tmapsto x(t,zeta)$ instead of $x(t)$, since the latter might be misunderstood).