|
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 a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses. Formal definition We consider a Markov process (Y_t)_ taking values in \mathcal. There is a measurable set \mathcal^of absorbing states and \mathcal^a = \mathcal \setminus \mathcal^. We denote by T the hitting time of \mathcal^, also called killing time. We denote by \ the family of distributions where \operatorname_x has original condition Y_0 = x \in \mathcal. We assume that \mathcal^ is almost surely reached, i.e. \forall x \in \mathcal, \operatorname_x(T t) = \nu(B)where \operatorname_\nu = \int_ \operatorname_x \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. 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 proc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing State
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Kernel
In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define random measures or stochastic processes. The most important example of kernels are the Markov kernels. Definition Let (S, \mathcal S) , (T, \mathcal T) be two measurable spaces. A function : \kappa \colon S \times \mathcal T \to , +\infty is called a (transition) kernel from S to T if the following two conditions hold: *For any fixed B \in \mathcal T , the mapping :: s \mapsto \kappa(s,B) :is \mathcal S/ \mathcal B( , +\infty-measurable; *For every fixed s \in S , the mapping :: B \mapsto \kappa(s, B) :is a measure on (T, \mathcal T). Classification of transition kernels Transition kernels are usually classified by the measures they define. Those measures are defined as : \kappa_s \colon \mathcal T \to , + \infty with : \kappa_s(B)=\kappa(s,B) for all B \in \mathcal T and all s \in S . With thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branching Process
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables. The random variables of a stochastic process are indexed by the natural numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation n produces some random number of individuals in generation n+1, according, in the simplest case, to a fixed probability distribution that does not vary from individual to individual. Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor. A central question in the the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-birth–death Process
In queueing models, a discipline within the mathematical theory of probability, the quasi-birth–death process describes a generalisation of the birth–death process. As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks. Discrete time The stochastic matrix describing the Markov chain has block structure ::P=\begin A_1^\ast & A_2^\ast \\ A_0^\ast & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& \ddots & \ddots & \ddots \end where each of ''A''0, ''A''1 and ''A''2 are matrices and ''A''*0, ''A''*1 and ''A''*2 are irregular matrices for the first and second levels. Continuous time The transition rate matrix for a quasi-birth-death process has a tridiagonal block structure ::Q=\begin B_ & B_ \\ B_ & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& A_0 & A_1 & A_2 \\ &&&& \ddots & \ddots & \ddots \end where each of ''B''00, '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voter Model
In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975. One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, for one of the chosen voter's neighbors is chosen according to a given set of probabilities and that individual's opinion is transferred to the chosen voter. An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
