Isserlis' Theorem
In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis. This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of . Other applications include the analysis of portfolio returns, quantum field theory and generation of colored noise. Statement If (X_1,\dots, X_) is a zero-mean multivariate normal random vector, then\operatorname ,X_1 X_2\cdots X_\,= \sum_\prod_ \operatorname ,X_i X_j\,= \sum_\prod_ \operatorname(\,X_i, X_j\,), where the sum is over all the pairings of \, i.e. all distinct ways of partitioning \ into pairs \, and the product is over the pairs contained in p. In his original paper, Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the 4^ order moments, which takes the appearance : ...
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Probability Theory
Probability theory 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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ...
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