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The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge. As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening.
The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
The word stochastic is used to describe other terms and objects in mathematics. Examples include a stochastic matrix, which describes a stochastic process known as a Markov process, and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process, also called the Brownian ...
This is an example of a covariance structure, many different types of which may be modeled in a random field. One example is the Ising model where sometimes nearest neighbor interactions are only included as a simplification to better understand the model.
In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero.
In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes.It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time.
In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in the formalization of random (stochastic) processes.