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If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
Many authors do not name this test or give it a shorter name. [2] When testing if a series converges or diverges, this test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality.
The only divergence for probabilities over a finite alphabet that is both an f-divergence and a Bregman divergence is the Kullback–Leibler divergence. [8] The squared Euclidean divergence is a Bregman divergence (corresponding to the function x 2 {\displaystyle x^{2}} ) but not an f -divergence.
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.
The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality: (,) ().One also has the following inequality, due to Bretagnolle and Huber [2] (see also [3]), which has the advantage of providing a non-vacuous bound even when () >:
Ball divergence is a non-parametric two-sample statistical test method in metric spaces. It measures the difference between two population probability distributions by integrating the difference over all balls in the space. [1] Therefore, its value is zero if and only if the two probability measures are the same.
A Stein discrepancy is a statistical divergence between two probability measures that is rooted in Stein's method.It was first formulated as a tool to assess the quality of Markov chain Monte Carlo samplers, [1] but has since been used in diverse settings in statistics, machine learning and computer science.