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The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and ...
The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope.Using the matrix entries as Cartesian coordinates, it lies in an ()-dimensional affine subspace of -dimensional Euclidean space defined by independent linear constraints specifying that the row and column sums all equal 1.
In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32] The word stochastic is used to describe other terms and objects in mathematics.
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
The theory of stochastic quantum mechanics is ascribed to Edward Nelson, who independently discovered a derivation of the Schrödinger equation within this framework. [1] [2] This theory was also developed by Davidson, Guerra, Ruggiero, Pavon and others. [7]
In probability theory, the matrix analytic method is a technique to compute the stationary probability ... An M/G/1-type stochastic matrix is one of the ...
In 1953, Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. [264] [263] Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer.
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.. The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable, which is the standard way of characterizing a discrete probability ...