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