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Proofs of convergence of random variables

This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {Xn} converges

Randomness

In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible

Exchangeable random variables

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distributi

Differential entropy

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy, a measure of averag

Index of dispersion

In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a

Information fluctuation complexity

Information fluctuation complexity is an information-theoretic quantity defined as the fluctuation of information about entropy. It is derivable from fluctuations in the predominance of order and chao

Random binary tree

In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Two different distributions are commonly used:

Alignments of random points

Alignments of random points in a plane can be demonstrated by statistics to be counter-intuitively easy to find when a large number of random points are marked on a bounded flat surface. This has been

Random element

In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet who comm

Randomness test

A randomness test (or test for randomness), in data evaluation, is a test used to analyze the distribution of a set of data to see if it can be described as random (patternless). In stochastic modelin

Clustering illusion

The clustering illusion is the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random. The illusion is caused by a

Statistical fluctuations

Statistical fluctuations are fluctuations in quantities derived from many identical random processes. They are fundamental and unavoidable. It can be proved that the relative fluctuations reduce as th

Infinite monkey theorem

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of Wi

Seven states of randomness

The seven states of randomness in probability theory, fractals and risk analysis are extensions of the concept of randomness as modeled by the normal distribution. These seven states were first introd

Random compact set

In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.

Random sequence

The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin

Poisson random measure

Let be some measure space with -finite measure . The Poisson random measure with intensity measure is a family of random variables defined on some probability space such that i) is a Poisson random va

Gaussian process emulator

In statistics, Gaussian process emulator is one name for a general type of statistical model that has been used in contexts where the problem is to make maximum use of the outputs of a complicated (of

Random variate

In probability and statistics, a random variate or simply variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable might have diff

Entropy estimation

In various science/engineering applications, such as independent component analysis, image analysis, genetic analysis, speech recognition, manifold learning, and time delay estimation it is useful to

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a

Statistical randomness

A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statisti

Concentration dimension

In mathematics — specifically, in probability theory — the concentration dimension of a Banach space-valued random variable is a numerical measure of how "spread out" the random variable is compared t

Control variates

The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate

Entropy (information theory)

In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variab

Stochastic computing

Stochastic computing is a collection of techniques that represent continuous values by streams of random bits. Complex computations can then be computed by simple bit-wise operations on the streams. S

Complete spatial randomness

Complete spatial randomness (CSR) describes a point process whereby point events occur within a given study area in a completely random fashion. It is synonymous with a homogeneous spatial Poisson pro

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