Stochastic Process

A stochastic process, [math]\displaystyle{ X_{t \in \mathcal{T}} }[/math], is a random element that is a process (whose outcomes cannot be foretold, but do have some statistical properties).

• AKA: Non-Deterministic Random Process, Probabilistic System.
• Context:
  • It can be modeled by a Stochastic Process Model.
  • It can have a Stochastic Process Property (such as a Markov process property).
  • It can range from being an Discrete-Outcome Stochastic Process to being a Continuous-Outcome Stochastic Process.
  • It can range from being an Discrete-Time Stochastic Process to being a Continuous-Time Stochastic Process.
  • It can range from being a Stable Stochastic Process to being a Nonstable Stochastic Process.
  • It can range from being a Repeatable Stochastic Process to being an Unrepeatable Stochastic Process.
  • It can range from being an Observable Stochastic Process to being an Unobservable Stochastic Process.
  • It can range from being a Mixture Stochastic Process to being a Single Stochastic Process.
  • It can range from being a Stationary Stochastic Process to being a Non-Stationary Stochastic Process (dynamic stochastic process).
  • It can be a Sequential Stochastic Process, such as a temporal stochastic process.
  • It can be represented by a Probability Function (from some statistical metamodel such as a temporal Bayesian network).
  • ...
• Example(s):
  • a Random Experiment (i.e. a stable repeatable observable stochastic process with a predefined outcome set), such as a Coin Toss Experiment.
  • a Random Walk.
  • a Markov Process.
  • a Poisson Process.
  • a Gaussian Process.
  • a Randomized Experiment-Subject Selection Process.
  • …
• Counter-Example(s):
  • a Deterministic Process.
  • a Random-Controlled Experiment.
  • a Statistical Model.
• See: Noise, Error, Stochastic Process Prior, Data-Generating Process, Determinantal Point Process, Stochastic Calculus, Ergodicity.

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References

2015a

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Random_element#Random_process
  • A Random process is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
    In the simple case of discrete time, as opposed to continuous time, a stochastic process involves a sequence of random variables and the time series associated with these random variables (for example, see Markov chain, also known as discrete-time Markov chain).

2015b

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/stochastic_process Retrieved:2015-2-16.
  • In probability theory, a stochastic process, or sometimes random process (widely used) is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

    In the simple case of discrete time, as opposed to continuous time, a stochastic process involves a sequence of random variables and the time series associated with these random variables (for example, see Markov chain, also known as discrete-time Markov chain). One approach to stochastic processes treats them as functions of one or several deterministic arguments (inputs; in most cases this will be the time parameter) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all take values in the same space (the codomain of the function). Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.

    Familiar examples of processes modeled as stochastic time series include stock market and exchange rate fluctuations, signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature, and random movement such as Brownian motion or random walks. Examples of random fields include static images, random terrain (landscapes), wind waves or composition variations of a heterogeneous material.

    A generalization, the random field, is defined by letting the variables' parameters be members of a topological space instead of limited to real values representing time.

2010

• (Riddles, 2010) ⇒ John Riddles. (2010). “Introduction to Stochastic Processes." Unpublished Glossary. Department of Statistics, Iowa State University.
  • QUOTE:
    • Stochastic Process: A Random Variable indexed by time (for an infinite number of time points). Equivalently, it is a Random Vector of infinite dimension.

      State Space: X(t) has a support for each value of t. The state space is the union of all of these supports.

2009a

• (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn
  • Stochastic Process: a statistical process involving a number of random variables depending on a variable parameter (which is usually time)

2009b

• (Encyclopaedia Britannica, 2009) ⇒ http://www.britannica.com/EBchecked/topic/285198/Index
  • Stochastic Process: in probability theory, a process involving the operation of chance. For example, in radioactive decay every atom is subject to a fixed probability of breaking down in any given time interval. More generally, a stochastic process refers to a family of random variables indexed against some other variable or set of variables. It is one of the most general objects of study in probability. Some basic types of stochastic processes include Markov process es, Poisson processes (such as radioactive decay), and time series, with the index variable referring to time. This indexing can be either discrete or continuous, the interest being in the nature of changes of the variables with respect to time.

2009c

• (Unknown, 2009) ⇒ http://www.dg.dial.pipex.com/documents/docs1/bullock29.shtml
  • Stochastic Process: Any process governed by the laws of probability. In the case of the language process, the expected probability of occurrence of any single element ...

2009d

• (AMS Glossary, 2009) ⇒ http://amsglossary.allenpress.com/glossary/browse
  • Stochastic Process: A system that evolves in time according to probabilistic equations, that is, the behavior of the system is determined by one or more time-dependent random variables.

2009e

• (Nature, 2009) ⇒ http://www.nature.com/nrg/journal/v3/n5/glossary/nrg795_glossary.html
  • Stochastic Process: A mathematical description of the random evolution of a quantity through time.

2009f

• (US NIH, 2009) ⇒ US National Institute of Health. “IUPAC Glossary of Terms Used in Toxicology."
  • stochastic: Pertaining to or arising from chance and hence obeying the laws of probability.
  • stochastic effect: Phenomenon pertaining to or arising from chance, and hence obeying the laws of probability.
  • stochastic process: See stochastic effect

2007

• (Watkins, 2007) ⇒ Joseph C. Watkins. (2007). “Discrete Time Stochastic Processes."
  • QUOTE: A stochastic process X (or a random process, or simply a process) with index set $\Lambda$ and a measurable state space (S, B) defined on a probability space $(\Omega,F, P)$ is a function
    [math]\displaystyle{ X : \Lambda \times \Omega \rightarrow S }[/math]

    such that for each $\lambda \in \Lambda$ ,

    [math]\displaystyle{ X(\lambda, \cdot) : \Omega \rightarrow S }[/math]

    is an S-valued random variable. Note that $\Lambda$ is not given the structure of a measure space. In particular, it is not necessarily the case that $X$ is measurable. However, if $\Lambda$ is countable and has the power set as its $\sigma$-algebra, then $X$ is automatically measurable.

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