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Harris Chain
In the mathematical study of stochastic processes, a Harris chain is a Markov chain where the chain returns to a particular part of the state space an unbounded number of times. Harris chains are regenerative processes and are named after Theodore Harris. The theory of Harris chains and Harris recurrence is useful for treating Markov chains on general (possibly uncountably infinite) state spaces. Definition Let \ be a Markov chain on a general state space \Omega with stochastic kernel K. The kernel represents a generalized one-step transition probability law, so that P(X_\in C\mid X_n=x)=K(x,C) for all states x in \Omega and all measurable sets C\subseteq \Omega. The chain \ is a ''Harris chain''R. Durrett. ''Probability: Theory and Examples''. Thomson, 2005. . if there exists A\subseteq\Omega,\varepsilon>0, and probability measure \rho with \rho(\Omega)=1 such that # If \tau_A:=\inf \, then P(\tau_A 0, and let ''A'' and Ω be open sets containing ''x''0 and ''y''0 respec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic 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 process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
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 dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. This property can be used in the derivation of theoretical properties of such processes. History Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955. Definition A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself. These time point may themselves be determined by the evolution of the process. That is to say, the process is a regenerative process if there exist time points 0 ≤ ''T''0 < ''T''1 < ''T''2 < ... such that the post-''Tk'' process * has the same distribution as the post-''T''0 process * is independent of the pre-''Tk'' process for ''k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ted Harris (mathematician)
Theodore Edward Harris (11 January 1919 – 3 November 2005) was an American mathematician known for his research on stochastic processes, including such areas as general state-space Markov chains (often now called Harris chains), the theory of branching processes and stochastic models of interacting particle systems such as the contact process. The Harris inequality in statistical physics and percolation theory is named after him. He received his Ph.D. at Princeton University in 1947 under advisor Samuel Wilks. From 1947 until 1966 he worked for the RAND Corporation, heading their mathematics department from 1959 to 1965. From 1966 until retirement in 1989 he was Professor of Mathematics and Electrical Engineering at University of Southern California. He was elected to the United States National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Kernel
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. Formal definition Let (X,\mathcal A) and (Y,\mathcal B) be measurable spaces. A ''Markov kernel'' with source (X,\mathcal A) and target (Y,\mathcal B) is a map \kappa : \mathcal B \times X \to ,1/math> with the following properties: # For every (fixed) B \in \mathcal B, the map x \mapsto \kappa(B, x) is \mathcal A-measurable # For every (fixed) x \in X, the map B \mapsto \kappa(B, x) is a probability measure on (Y, \mathcal B) In other words it associates to each point x \in X a probability measure \kappa(dy, x): B \mapsto \kappa(B, x) on (Y,\mathcal B) such that, for every measurable set B\in\mathcal B, the map x\mapsto \kappa(B, x) is measurable with respect to the \sigma-algebra \mathcal A. Examples Simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Kernel
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f(t) = \int_^ (Tf)(u) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : ''absolutely continuous'' ⊆ ''uniformly continuous'' = ''cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Sets
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
