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Introduction

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The statement of the theorem

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or nn -dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is nn -dimensional Euclidean space, then the stochastic process is called a nn -dimensional vector process or nn -vector process. The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931. According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888. The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. The terms random process and stochastic process are considered synonyms and are used interchangeably, without the index set being precisely specified. Both "collection", or "family" are used while instead of "index set", sometimes the terms "parameter set" or "parameter space" are used. The term random function is also used to refer to a stochastic or random process, though sometimes it is only used when the stochastic process takes real values. This term is also used when the index sets are mathematical spaces other than the real line, while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is nn -dimensional Euclidean space Rn\mathbb {R} ^{n} or a manifold. A stochastic process can be denoted, among other ways, by {X(t)}t∈T\{X(t)\}_{t\in T} , {Xt}t∈T\{X_{t}\}_{t\in T} , {Xt}\{X_{t}\} {X(t)}\{X(t)\} or simply as XX . Some authors mistakenly write X(t)X(t) even though it is an abuse of function notation. For example, X(t)X(t) or XtX_{t} are used to refer to the random variable with the index tt , and not the entire stochastic process. If the index set is T=[0,∞)T=[0,\infty ) , then one can write, for example, (Xt,t≥0)(X_{t},t\geq 0) to denote the stochastic process. - ^ ^{a}^{b}Cite error: The named reference Parzen1999 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}^{e}^{f}Cite error: The named reference GikhmanSkorokhod1969page1 was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference doob1953stochasticP46to47 was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference KarlinTaylor2012page27 was invoked but never defined (see the help page). - ^Cite error: The named reference Applebaum2004page1337 was invoked but never defined (see the help page). - ^ ^{a}^{b}^{c}^{d}^{e}Cite error: The named reference Lamperti1977page1 was invoked but never defined (see the help page). - ^Cite error: The named reference RogersWilliams2000page121b was invoked but never defined (see the help page). - ^ ^{a}^{b}Cite error: The named reference Florescu2014page294 was invoked but never defined (see the help page). - ^ ^{a}^{b}Samuel Karlin; Howard E. 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ISBN978-0-19-856814-8. - ^Murray Rosenblatt (1962). Random Processes. Oxford University Press. p. 91. - ^John A. Gubner (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. p. 383. ISBN978-1-139-45717-0. - ^ ^{a}^{b}Kiyosi Itō (2006). Essentials of Stochastic Processes. American Mathematical Soc. p. 13. ISBN978-0-8218-3898-3. - ^M. Loève (1978). Probability Theory II. Springer Science & Business Media. p. 163. ISBN978-0-387-90262-3. - ^Pierre Brémaud (2014). Fourier Analysis and Stochastic Processes. Springer. p. 133. ISBN978-3-319-09590-5. - ^ ^{a}^{b}Gusak et al. (2010), p. 1 - ^Richard F. Bass (2011). Stochastic Processes. Cambridge University Press. p. 1. ISBN978-1-139-50147-7. - ^ ^{a}^{b},John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. p. 3. ISBN978-3-540-90275-1. - ^Fima C. Klebaner (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press. p. 55. 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