A transition matrix is called doubly stochastic if both its...

Name: Problem 19, Chapter 9: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-03-10T16:28:43-08:00
Duration: 3 min 59 s
Channel: Madi Sousa
Description: A transition matrix is called doubly stochastic if both its row and column sums are equal to 1 . What is the limiting probability state of a Markov chain with doubly stochastic transition matrix?

A transition matrix is called doubly stochastic if both its row and column sums are equal to 1 . What is the limiting probability state of a Markov chain with doubly stochastic transition matrix?

Key Concepts

Markov Chain

A Markov chain is a stochastic process that undergoes transitions from one state to another on a state space. It is characterized by the Markov property, which means that the future state depends only on the current state and not on the sequence of events that preceded it.

Transition Matrix

The transition matrix of a Markov chain is a square matrix that describes the probabilities of moving from any state to any other state. Each element of the matrix indicates the probability of transitioning from one state to another, with the rows summing to 1.

Doubly Stochastic Matrix

A doubly stochastic matrix is a special kind of transition matrix where, in addition to each row summing to 1, each column also sums to 1. This property implies a certain symmetry in the transition probabilities, which often leads to uniform behavior in long-term dynamics.

Stationary Distribution

The stationary distribution of a Markov chain is a probability distribution that remains unchanged as the system evolves over time. For any transition matrix, it satisfies the condition that when it is multiplied by the matrix, the result is the same distribution. In the case of a doubly stochastic matrix, the stationary distribution is often uniform.

Limiting Probability State

The limiting probability state refers to the distribution that the states of the Markov chain converge to as the number of transitions goes to infinity. For a Markov chain with a doubly stochastic transition matrix, assuming the chain is irreducible and aperiodic, this limiting distribution is the uniform distribution across all states.

Transcript

00:01 This is a very interesting problem because we are talking about the eigenvalues and eigenvectors of a matrix, and specifically we are relating these eigenvalues to the concept of probability, and more specifically, markov chains.

00:17 Now, this is a very involved problem.

00:20 There are multiple things to consider.

00:22 So let's just start by understanding what we're given in this problem.

00:28 So what are we given? we're given that we have a transition matrix, and we know that the rows and columns of this matrix add up to 1.

00:38 We're also told that a is an end -by -end doubly stochastic matrix whose eigenvalues satisfy a specific property.

00:47 We know that lambda 1 equals 1, and lambda j is less than 1 for j being some integer, 2, 3, all the way up to n.

00:56 But notice here that j is not 1.

01:00 So what are we doing in this problem? we're told that we have a vector which we called e in rn, whose entries equal 1, who add up to 1, then can we find and prove that the markov chain will converge to a value x equal to 1 over n times our vector e? first, let's understand what our transition matrix a would look like.

01:28 Remember that the columns and rows add to one.

01:32 So this is something that the matrix could look like.

01:34 For example, in this first column here, we have 0 .8, 0 .1, 0 .05, 0 .05.

01:41 They add to 1.

01:42 Now, if we rearrange these in the columns and check that the rows all add to 1, this is something that could satisfy the eigenvalues that we're given.

01:52 So then we can say that a and our transpose matrix a are stochastic.

01:58 Now, since the rows are doubly sarcastic in a and they add to 1, we know that e is an eigenvector of a whose eigenvalue is when lambda equals 1...

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