What is a doubly stochastic process?

What is a doubly stochastic process?

The doubly stochastic Poisson process (DSPP) is a generalization of the Poisson process when the intensity of the occurrence of the points is influenced by an external process called information process such that the intensity becomes a random process. This process was introduced by Cox [1].

Is doubly stochastic matrix symmetric?

A real symmetric matrix with non-negative entries with row sums and column sums equal to 1 is called doubly stochastic matrix. From the Perron–Frobenius theorem an eigenvector corresponding to is such that each of its entries are non-negative and sums to 1.

Is a doubly stochastic matrix invertible?

Can the inverse of a stochastic matrix be stochastic? Yes, if the matrix is doubly stochastic and orthogonal. In that case, the inverse is the transpose and still stochastic.

When a Markov matrix is called doubly stochastic?

Discrete-Time Markov Chains A transition probability matrix P is defined to be a doubly stochastic matrix if each of its columns sums to 1. That is, not only does each row sum to 1, each column also sums to 1. Thus, for every column j of a doubly stochastic matrix, we have that ∑ i p i j = 1 .

What is the limiting distribution for a Markov chain with a doubly stochastic regular transition probability matrix?

The Long Run Behavior of Markov Chains The matrix is doubly stochastic, and it is regular (P2 has only strictly positive entries), hence the limiting distribution is π = ( 1 7 , … , 1 7 ) .

What is sinkhorn Knopp algorithm?

The Sinkhorn-Knopp algorithm takes a matrix A and finds diagonal matrices D and E such that if M = DAE the sum of each column and each row of M is unity. The method is, in effect, to alternately normalise the rows and the columns of the matrix. Such matrices have various applications, including web page ranking.

What is transition matrix in Markov chain?

In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix.

How do you know if a stochastic matrix is regular?

A stochastic square matrix is regular if some positive power has all entries nonzero. If the transition matrix M for a Markov chain is regular, then the Markov chain has a unique limit vector (known as a steady-state vector), regardless of the values of the initial probability vector.

What does it mean for a stochastic matrix to be regular?

A stochastic matrix A is said to be regular if all elements of at least one particular power of A are positive and different from zero. Regular matrices are important for the calculation of probabilities of dependant processes (Markov chains).

Is a doubly stochastic matrix irreducible?

It can be seen that each row of the matrix sums to 1, and each column also sums to 1; that is, it is a doubly stochastic matrix. Because the process is an irreducible, aperiodic Markov chain, the limiting-state probabilities exist and are given by π 1 = π 2 = π 3 = 1 / 3 .

What is a stationary Markov chain?

A stationary distribution of a Markov chain is a probability distribution that remains unchanged in the Markov chain as time progresses. Typically, it is represented as a row vector π whose entries are probabilities summing to 1, and given transition matrix P, it satisfies.

What does stochastic matrix mean?

In mathematics, a stochastic matrix is a matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability.

What is stochastic differential equation?

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations.

What does stochastic variable mean?

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.