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A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right ...
The above matrix as a graph. (P) i j is the probability that, if a given day is of type i, it will be followed by a day of type j. Notice that the rows of P sum to 1: this is because P is a stochastic matrix. [4]
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 motion process.
Multiplying together stochastic matrices always yields another stochastic matrix, so Q must be a stochastic matrix (see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that Q is a stochastic matrix to solve for Q .
A Markov chain can be described by a stochastic matrix, which lists the probabilities of moving to each state from any individual state. From this matrix, the probability of being in a particular state n steps in the future can be calculated. A Markov chain's state space can be partitioned into communicating classes that describe which states ...
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.
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse, [21] and the Poisson process, used by A. K. Erlang to study the number of ...