An n ×n matrix P with entries Pi, j is called stochastic if...

An $n \times n$ matrix P with entries $P_{i, j}$ is called stochastic if all entries are nonnegative and if the sum of the entries in each row is 1 . It is called doubly stochastic if, additionally, the sum of the entries in each column is 1 . Show that the uniform distribution is a stationary distribution for any Markov chain represented by a donbly stochastic matrix.

Key Concepts

Stochastic Matrix

A stochastic matrix is a square matrix used to represent the transitions of a Markov chain, where every entry is nonnegative and the entries in each row sum to one. This structure ensures that the matrix behaves like a set of probability distributions, each corresponding to the next state probabilities from a given current state.

Doubly Stochastic Matrix

A doubly stochastic matrix extends the concept of a stochastic matrix by requiring, in addition to the rows summing to one, that each column also sums to one. This symmetric property means that the probability flow is balanced in both the forward and backward directions across the states.

Markov Chain

A Markov chain is a mathematical system that undergoes transitions from one state to another in discrete time steps, with the defining property that the next state depends only on the current state (the Markov property). Markov chains are typically represented by stochastic matrices that encapsulate the transition probabilities between states.

Stationary Distribution

A stationary distribution for a Markov chain is a probability distribution that remains unchanged as the system evolves over time. When a Markov chain reaches its stationary distribution, the distribution of states will remain constant after subsequent transitions, regardless of the initial distribution.

Uniform Distribution

The uniform distribution assigns equal probability to every state in the state space. In the context of doubly stochastic matrices, the uniform distribution is significant because it is a stationary distribution; applying a doubly stochastic matrix to a uniform distribution yields the same uniform distribution, maintaining equal probability across all states.

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