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Convergence of random variables

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept

Queueing theory

Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered

Kinetic scheme

In physics, chemistry and related fields, a kinetic scheme is a network of states and connections between them representing the scheme of a dynamical process. Usually a kinetic scheme represents a Mar

Cheeger bound

In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special

Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used

Girsanov theorem

In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how t

Moment closure

In probability theory, moment closure is an approximation method used to estimate moments of a stochastic process.

Long-tail traffic

A long-tailed or heavy-tailed probability distribution is one that assigns relatively high probabilities to regions far from the mean or median. A more formal mathematical definition is given below. I

Totally inaccessible stopping time

No description available.

Finite-dimensional distribution

In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process)

Filtration (mathematics)

In mathematics, a filtration is an indexed family of subobjects of a given algebraic structure , with the index running over some totally ordered index set , subject to the condition that if in , then

Feynman–Kac formula

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman we

Hierarchical Dirichlet process

In statistics and machine learning, the hierarchical Dirichlet process (HDP) is a nonparametric Bayesian approach to clustering grouped data. It uses a Dirichlet process for each group of data, with t

Law of the iterated logarithm

In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khin

Counting process

A counting process is a stochastic process {N(t), t ≥ 0} with values that are non-negative, integer, and non-decreasing: 1.
* N(t) ≥ 0. 2.
* N(t) is an integer. 3.
* If s ≤ t then N(s) ≤ N(t). If s

Bessel process

In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.

Stationary increments

In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many

Nuisance variable

In the theory of stochastic processes in probability theory and statistics, a nuisance variable is a random variable that is fundamental to the probabilistic model, but that is of no particular intere

Chinese restaurant process

In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant.Imagine a restaurant with an infinite number of ci

Predictable σ-algebra

No description available.

Adapted process

In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is th

Chernoff's distribution

In probability theory, Chernoff's distribution, named after Herman Chernoff, is the probability distribution of the random variable where W is a "two-sided" Wiener process (or two-sided "Brownian moti

Markov chain central limit theorem

In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theor

Stochastic control

Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the s

Intensity of counting processes

The intensity of a counting process is a measure of the rate of change of its predictable part. If a stochastic process is a counting process, then it is a submartingale, and in particular its Doob-Me

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smal

Frequency of exceedance

The frequency of exceedance, sometimes called the annual rate of exceedance, is the frequency with which a random process exceeds some critical value. Typically, the critical value is far from the mea

Local time (mathematics)

In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particl

Sample-continuous process

In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.

Two-state trajectory

A two-state trajectory (also termed two-state time trajectory or a trajectory with two states) is a dynamical signal that fluctuates between two distinct values: ON and OFF, open and closed, , etc. Ma

Information source (mathematics)

In mathematics, an information source is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution. The uncertainty, or entropy rate, of an information source i

Ruin theory

In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such

Quadratic variation

In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.

Rough path

In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregul

Active Brownian particle

An active Brownian particle (ABP) is a model of self-propelled motion in a dissipative environment. It is a nonequilibrium generalization of a Brownian particle. The self-propulsion results from a for

Lorden's inequality

In probability theory, Lorden's inequality is a bound for the moments of overshoot for a stopped sum of random variables, first published by Gary Lorden in 1970. Overshoots play a central role in rene

Σ-Algebra of τ-past

The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probabi

Oscillator linewidth

The concept of a linewidth is borrowed from laser spectroscopy. The linewidth of a laser is a measure of its phase noise. The spectrogram of a laser is produced by passing its light through a prism. T

List of stochastic processes topics

In the mathematics of probability, a stochastic process is a random function. In practical applications, the domain over which the function is defined is a time interval (time series) or a region of s

Hitting time

In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times

Risk of ruin

Risk of ruin is a concept in gambling, insurance, and finance relating to the likelihood of losing all one's investment capital or extinguishing one's bankroll below the minimum for further play. For

Numéraire

The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradabl

Random measure

In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such

Stochastic simulation

A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities. Realizations of these random variables are generated and

Single-particle trajectory

Single-particle trajectories (SPTs) consist of a collection of successive discrete points causal in time. These trajectories are acquired from images in experimental data. In the context of cell biolo

Jump process

A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson

Regenerative process

In applied probability, a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated as being statistically independent of each other.

Pitman–Yor process

In probability theory, a Pitman–Yor process denoted PY(d, θ, G0), is a stochastic process whose sample path is a probability distribution. A random sample from this process is an infinite discrete pro

Extinction probability

Extinction probability is the chance of an inherited trait becoming extinct as a function of time t. If t = ∞ this may be the complement of the chance of becoming a universal trait.
* v
* t
* e

Recurrence period density entropy

Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal

Supersymmetric theory of stochastic dynamics

Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicabili

Random walk

In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random w

Fokker–Planck equation

In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the in

Continuous stochastic process

In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property

Progressively measurable process

In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the

Natural filtration

In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which r

Stopped process

In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time.

Dissociated press

Dissociated press is a parody generator (a computer program that generates nonsensical text). The generated text is based on another text using the Markov chain technique. The name is a play on "Assoc

Sazonov's theorem

In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov (Вячесла́в Васи́льевич Сазо́нов), is a theorem in functional analysis. It states that a bounded linear operator between tw

Abstract Wiener space

The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emph

Increasing process

An increasing process is a stochastic process where the random variables which make up the process are increasing almost surely and adapted: A continuous increasing process is such a process where the

Spherical contact distribution function

In probability and statistics, a spherical contact distribution function, first contact distribution function, or empty space function is a mathematical function that is defined in relation to mathema

Stopping time

In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time) is a specific type of “random time

Ionescu-Tulcea theorem

In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem deals with the existence of probability measures for probabilistic events c

Feller-continuous process

In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously

Borel right process

In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process. Let be a locally compact, separable, metric space.We

Gaussian measure

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimen

Jyotiprasad Medhi

Jyotiprasad Medhi was a professor of statistics at Gauhati University and Institute of Advanced Study in Science and Technology.

Stochastic homogenization

In homogenization theory, a branch of mathematics, stochastic homogenization is a technique for understanding solutions to partial differential equations with oscillatory random coefficients.

Continuity in probability

In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.

Random walk hypothesis

The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.

Gibbs state

In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system. For example, a stationary or ste

Stochastic process

In probability theory and related fields, a stochastic (/stoʊˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used

Accessible stopping time

No description available.

Spitzer's formula

In probability theory, Spitzer's formula or Spitzer's identity gives the joint distribution of partial sums and maximal partial sums of a collection of random variables. The result was first published

Excursion probability

In probability theory, an excursion probability is the probability that a stochastic process surpasses a given value in a fixed time period. It is the probability Numerous approximation methods for th

Jump diffusion

Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and

Stochastic resonance

Stochastic resonance (SR) is a phenomenon in which a signal that is normally too weak to be detected by a sensor, can be boosted by adding white noise to the signal, which contains a wide spectrum of

Martingale (probability theory)

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equ

Russo–Vallois integral

In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral for suitable functions and . The idea is to replace the derivat

Stochastic cellular automaton

Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains are an important extension of cellular automaton. Cellular automa

System size expansion

The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows on

Zero-order process (statistics)

In probability theory and statistics, a zero-order process is a stochastic process in which each observation is independent of all previous observations. For example, a zero-order process in marketing

Least-squares spectral analysis

Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos

Narrow escape problem

The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology. The mathematical formulation is the following: a Brownian particle (ion, molecule, or protein) is confine

Weakly dependent random variables

In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale. A (time) sequence of random variables is weakly dependent if di

Random Fibonacci sequence

In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation , where the signs + or − are chosen at random with equal probability

Moran process

A Moran process or Moran model is a simple stochastic process used in biology to describe finite populations. The process is named after Patrick Moran, who first proposed the model in 1958. It can be

Version (probability theory)

No description available.

Disorder problem

In the study of stochastic processes in mathematics, a disorder problem or quickest detection problem (formulated by Kolmogorov) is the problem of using ongoing observations of a stochastic process to

Basic affine jump diffusion

In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form where is a standard Brownian motion, and is an independent compound Poisson process w

Random dynamical system

In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are cha

Stationary process

In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distri

Poisson boundary

In mathematics, the Poisson boundary is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when

Dependent Dirichlet process

In the mathematical theory of probability, the dependent Dirichlet process (DDP) provides a non-parametric prior over evolving mixture models. A construction of the DDP built on a Poisson point proces

Quasi-stationary distribution

In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for

Subordinator (mathematics)

In probability theory, a Subordinator is a stochastic process that is non-negative and whose increments are stationary and independent. Subordinators are a special class of Lévy process that play an i

Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variable

Life-time of correlation

In probability theory and related fields, the life-time of correlation measures the timespan over which there is appreciable autocorrelation or cross-correlation in stochastic processes.

Gaussian free field

In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of

Càdlàg

In mathematics, a càdlàg (French: "continue à droite, limite à gauche"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function

Large deviations of Gaussian random functions

A random function – of either one variable (a random process), or two or more variables(a random field) – is called Gaussian if every finite-dimensional distribution is a multivariate normal distribut

Killed process

In probability theory — specifically, in stochastic analysis — a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time.

Kolmogorov's inequality

In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables ex

Stochastic geometry

In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, h

Hawkes process

In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process. It has arrivals at times where the infinitesimal probability of an arrival

Stochastic quantization

In theoretical physics, stochastic quantization is a method for modelling quantum mechanics, introduced by Edward Nelson in 1966, and streamlined by Parisi and Wu.

Gaussian noise

Gaussian noise, named after Carl Friedrich Gauss, is a term from signal processing theory denoting a kind of signal noise that has a probability density function (pdf) equal to that of the normal dist

Onsager–Machlup function

The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to th

Dirichlet process

In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions

Contact process (mathematics)

The contact process is a stochastic process used to model population growth on the set of sites of a graph in which occupied sites become vacant at a constant rate, while vacant sites become occupied

Law (stochastic processes)

In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. The law encodes a lot of information abo

Statistical fluctuations

Statistical fluctuations are fluctuations in quantities derived from many identical random processes. They are fundamental and unavoidable. It can be proved that the relative fluctuations reduce as th

Bernoulli process

In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes onl

Classical Wiener space

In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually n-dimensional

File dynamics

The term file dynamics is the motion of many particles in a narrow channel. In science: in chemistry, physics, mathematics and related fields, file dynamics (sometimes called, single file dynamics) is

Stochastic thermodynamics

Stochastic thermodynamics is an emergent field of research in statistical mechanics that uses stochastic variables to better understand the non-equilibrium dynamics present in many microscopic systems

Pregaussian class

In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian

Schramm–Loewner evolution

In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling li

Stochastic

Stochastic (/stəˈkæstɪk/, from Greek στόχος (stókhos) 'aim, guess') refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are disti

Continuous-time stochastic process

In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of

Link-centric preferential attachment

In mathematical modeling of social networks, link-centric preferential attachmentis a node's propensity to re-establish links to nodes it has previously been in contact with in time-varying networks.

Polynomial chaos

Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms of a polynomial function of other random variabl

Predictable stopping time

No description available.

Additive process

An additive process, in probability theory, is a cadlag, continuous in probability stochastic process with independent increments.An additive process is the generalization of a Lévy process (a Lévy pr

Resource-dependent branching process

A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal

Galton–Watson process

The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (p

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