Law of series/Stochastic process

Basic definitions

A stochastic process is a family of real random variables $(X_t)_{t\in T}$ defined on a probability space $(\Omega,\Sigma,P)$, where the set $T$ is interpreted as time. Time should have at least semigroup structure, usually it is either $\mathbb R$, $\mathbb R_{+\{0\}}$ (continuous time), $\mathbb Z$ or $\mathbb N$ (discrete time). Each $\omega\in \Omega$ determins a trajectory (or realization) of the process, i.e., the function $t\mapsto X_t(\omega)$. A process is considered to be defined by its joint distributions and so the particular space on which the variables $X_t$ are defined is not important and all that really matters is the distribution of the process. Thus one is free to choose the underlying space $(\Omega,\Sigma,P)$ as long as the joint distribution is left unchanged. The joint distribution of the process is determined by the finite-dimensional distributions, i.e., by the joint distributions of the tuples $X_{t_1}, X_{t_2},...,X_{t_k}$ for all $k\ge 1$ and all vectors $t_1, t_2 ,..., t_k$ of times.

Special types of processes are defined by additional properties. Some of them are listed below:

Stationary and homogeneous processes

A stochastic process whose time is a semigroup is stationary if for every $s\in T$ the process $Y_t=X_{t+s}$ has the same finite-dimensional distributions as $X_t$.

A similar condition defines homegeneous processes. For every $s\in T$ the process $Y_t=X_{t+s}-X_s$ has the same finite-dimensional distributions as $X_t$. For example, if $X_t$ is stationary and has integrable (with respect to the time) trajectories, then the integral process $Z_t$ defined for each $\omega$ and $t$ by $Z_t(\omega) = \int_0^t X_s(\omega) ds$ is homogeneous.

Signal processes

A signal process is a continuous time integer valued stochastic process with the following two properties: 1. $X_0 = 0$ almost surely, 2. the trajectories $t\mapsto X_t(\omega)$ are almost surely nondecreasing in $t$. Clearly, the trajectories must have discontinuities (jumps from one integer to a higher one). These jumps are interpreted as signals and then $X_t$ counts the number of signals in the time interval $[0,t]$.

Homogeneous signal processes are an important class and an example of such a process is the Poisson process. For a homogeneous signal process, the waiting time is the random variable defined on $\Omega$ as the time of the first signal after time 0:

$$V(\omega) = \inf\{t: X_t(\omega)\ge 1\}.$$ The distribution function $F$ of $V$ depends only on the one-dimensional distributions of the process, namely, we have $$F(t) = P\{V\le t\} = P\{X_t\ge 1\} = 1 - P\{X_t = 0\}.$$

See also

Wikipedia: Stochastic process]