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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. Definition Let X=(X_t)_ be a stochastic process in \R^n . The process X is continuous in probability when X_r converges in probability to X_s whenever r converges to s . Examples and Applications Feller processes are continuous in probability at t=0 . Continuity in probability is a sometimes used as one of the defining property for Lévy process. Any process that is continuous in probability and has independent increments has a version that is càdlàg In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous an .... As a result, some authors immediately define Lévy process as being càdlàg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...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 in a probability space, where the index of the family often has the interpretation of time. 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 Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution. The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller Process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process. Definitions Let ''X'' be a locally compact Hausdorff space with a countable base. Let ''C''0(''X'') denote the space of all real-valued continuous functions on ''X'' that vanish at infinity, equipped with the sup-norm , , ''f'' , , . From analysis, we know that ''C''0(''X'') with the sup norm is a Banach space. A Feller semigroup on ''C''0(''X'') is a collection ''t'' ≥ 0 of positive linear maps from ''C''0(''X'') to itself such that * , , ''T''''t''''f'' , , ≤ , , ''f'' , , for all ''t'' ≥ 0 and ''f'' in ''C''0(''X''), i.e., it is a contraction (in the weak sense); * the semigroup property: ''T''''t'' + ''s'' = ''T''''t'' \circ''T''''s'' for all ''s'', ''t'' ≥ 0; * lim''t'' → 0, , ''T''''t''''f'' − ''f'' , , = 0 for every ''f'' in ''C''0(''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths. All Lévy processes are additive processes. Mathematical definition A Lévy process is a stochastic process X=\ that satisfi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Increments
In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochastic processes that by definition possess independent increments are the Wiener process, all Lévy processes, all additive process and the Poisson point process. Definition for stochastic processes Let (X_t)_ be a stochastic process. In most cases, T= \N or T=\R^+ . Then the stochastic process has independent increments if and only if for every m \in \N and any choice t_0, t_1, t_2, \dots,t_, t_m \in T with : t_0 < t_1 < t_2< \dots < t_m the : are [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Version (probability Theory)
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables in a probability space, where the Index set, index of the family often has the interpretation of time. 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, stochastic control, control theory, information theory, computer science, 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 proposa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Càdlàg
In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space. Two related terms are càglàd, standing for "", the left-right reversal of càdlàg, and càllàl for "" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. Definition Let (M, d) be a metric space, and let E \subseteq \mathbb. A function f:E \to M is called a càdlàg function if, for every t \in E, * the left limit f(t-) := \lim_f(s) exists; and * the right limit f(t+) := \lim_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
