# Confusion in definition of Stochastic Process - Papoulis 4th Ed

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.

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).

• I have edited my question for clarity. My doubt is that the definition is very restrictive. Mar 16, 2019 at 13:34
• "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.
– Will
Mar 16, 2019 at 14:22