Taylor, Jonathan E. Random fields and the geometry of Wiener space. More by Jonathan E. Abstract Article info and citation First page References Abstract In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. Article information Source Ann. Export citation. Export Cancel. In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time , such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise , or the movement of a gas molecule.
They have applications in many disciplines including sciences such as biology ,  chemistry ,  ecology ,  neuroscience ,  and physics  as well as technology and engineering fields such as image processing , signal processing ,  information theory ,  computer science ,  cryptography  and telecommunications. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes.
Examples of such stochastic processes include the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse ,  and the Poisson process , used by A. Erlang to study the number of phone calls occurring in a certain period of time. The term random function is also used to refer to a stochastic or random process,   because a stochastic process can also be interpreted as a random element in a function space.
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. Historically, the index set was some subset of the real line , such as the natural numbers , giving the index set the interpretation of time.
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.
The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. 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.
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 as its earliest occurrence. The term stochastic process first appeared in English in a paper by Joseph Doob. 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 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 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. Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.
A classic example of a random walk is known as the simple random walk , which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.
Almost surely , a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk.
The Poisson process is a stochastic process that has different forms and definitions. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set.
This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.
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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. Defined on the real line, the Poisson process can be interpreted as a stochastic process,   among other random objects. There are others ways to consider a stochastic process, with the above definition being considered the traditional one. The state space is defined using elements that reflect the different values that the stochastic process can take.
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A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. The law of a stochastic process or a random variable is also called the probability law , probability distribution , or the distribution.
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The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions. Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed.
The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity.
A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such in the case of the index set being some subset of the real numbers. A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. Two stochastic processes that are modifications of each other have the same law  and they are said to be stochastically equivalent or equivalent.
Instead of modification, the term version is also used,     however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.
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If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. Separability is a property of a stochastic process based on its index set in relation to the probability measure.
The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space , [b] which means that the index set has a dense countable subset. The concept of separability of a stochastic process was introduced by Joseph Doob ,  where the underlying idea is to make a countable set of points of the index set determine the properties of the stochastic process.
Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process on the real line , are also members of this space. In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.
Markov processes are stochastic processes, traditionally in discrete or continuous time , that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process. The Brownian motion process and the Poisson process in one dimension are both examples of Markov processes  in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.
A Markov chain is a type of Markov process that has either discrete state space or discrete index set often representing time , but the precise definition of a Markov chain varies. Markov processes form an important class of stochastic processes and have applications in many areas. A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value.
In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,    but they can also be complex-valued  or even more general.
A symmetric random walk and a Wiener process with zero drift are both examples of martingales, respectively, in discrete and continuous time. Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process on the real line resulting in a martingale called the compensated Poisson process.
Martingales mathematically formalize the idea of a fair game,  and they were originally developed to show that it is not possible to win a fair game.
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field.
Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,   but very little analysis on them was done in terms of probability. After Cardano, Jakob Bernoulli [e] wrote Ars Conjectandi , which is considered a significant event in the history of probability theory.