# Law of series/Stochastic process

## Contents

### 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 $F(t) = P\{V\le t\} = P\{X_t\ge 1\} = 1 - P\{X_t = 0\}.$