Stochastic Process

It's helpful to define a stochastic process via an analogy to RVs. An RV $$\mathbf x$$ is a rule for assigning to every outcome $$\zeta$$ a number $$\mathbf x(\zeta)$$. A stochastic process is a rule for assigning to every outcome $$\zeta$$ a function $$\mathbf x(t, \zeta)$$. Thus a stochastic process is a family of time functions depending on the parameter $$\zeta$$ or, equivalently, a function of $$t$$ and $$\zeta$$. The domain of $$\zeta$$ is the set of all experimental outcomes and the domain of $$t$$ is a set $$R$$ of real numbers. (Note: It's not the set of real numbers, it's a set of real numbers.

If $$R$$ is the real axis, then $$\mathbf x(t)$$ is a continuous-time process. If $$R$$ is the set of integers, then $$\mathbf x(t)$$ is a discrete-time process. A discrete-time process is, thus, a sequence of RVs. Such a sequence is typically denoted by $$\mathbf x_n$$ or $$\mathbf x[n]$$.

The symbol $$\mathbf x(t)$$ has multiple interpretations:


 * It's a family of functions $$\mathbf x(t,\zeta)$$. In this interpretation, $$t$$ and $$\zeta$$ are variables.
 * It's 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 $$\mathbf x(t)$$ is an RV.
 * If $$t$$ and $$\zeta$$ are fixed, then $$x(t)$$ is a number.

Discrete stochastic processes
A probably even simpler definition can be given as below. First of all, note that random processes are infinite. Consider the coin tossing experiment. Then, there are an infinite number of infinite long possible experimental outcomes,

$$\mathcal S = \{(H, H, T, \dots), (H, T, H, \dots), (T, H, H, \dots), (T, T, T, \dots), \dots\}$$.

Recall that a random variable is simply a mapping between all possible experimental outcomes and a numerical value. Similarly, a random process is a mapping that maps every possible outcome in $$\mathcal S$$ to a series of numbers. That is,

$$\mathcal S_{\mathcal X} = {(1,1,0, \dots), (1,0,1, \dots), (0,1,1, \dots), (0,0,0, \dots}), \dots$$.

The random process does nothing but mapping each sequence in $$\mathcal S$$ to a sequence in $$\mathcal S_{\mathcal X}$$. Again, there is nothing random about this procedure, it's a pretty deterministic mapping and the name random process is somewhat unfortunate.

According to time resolution and output type

 * Discrete time discrete valued (DTDV)
 * Discrete time continuous valued (DTCV)
 * Continuous time discrete  valued (CTDV)
 * Continuous time continuous valued (CTCV)

According to time dependence
Random processes can be categorized from simplest to most complicated (in terms of time dependence) as below
 * IID processes (independent and identically distributed)
 * Stationary processes
 * Wide-sense stationary processes

The lower items in this list encompass the higher items (i.e. a stationary process is also IID etc.)