Are Markov chains IID?

Are Markov chains IID?

Any iid sequence forms a Markov chain, for if {Xn} is iid, then {Xn+1,Xn+2,…} In fact {Xn+1,Xn+2,…} is independent of {X0,…,Xn} (the past and the present): For an iid sequence, the future is independent of the past and the present state.

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.

Is a sequence of independent random variables a Markov chain?

Let {Xn} be a sequence of independent random variables. Then, {Xn} is a Markov chain (The Markov property holds here trivially since the past does not influence the future at all). It is stationary if and only if the variables have the same distribution . + Xn} is a Markov chain.

Is the stationary distribution of a Markov chain unique?

When there is only one equivalence class we say the Markov chain is irreducible. We will show that for an irreducible Markov chain, a stationary distri- bution exists if and only if all states are positive recurrent, and in this case the stationary distribution is unique.

How are random variables used in a Markov chain?

Extension of the Bootstrap methods from i.i.d. random variables to Markov chains was initiated by Kulperger and Prakasa Rao (1989) for the finite state space case. Suppose that {Xn}n ≥ 0 be a stationary Markov chain with a finite state space S = {s1, …, sℓ}, where ℓ∈ℕ and where ℕ ≡ {1, 2, …} denotes the set of all natural integers.

What’s the difference between ergodic and absorbing Markov chains?

Ergodic Markov chains have a unique stationary distribution, and absorbing Markov chains have stationary distributions with nonzero elements only in absorbing states. The stationary distribution gives information about the stability of a random process and, in certain cases, describes the limiting behavior of the Markov chain.

When does a Markov chain have multiple communicating classes?

However, this can only occur when the Markov chain is reducible, i.e. has multiple communicating classes. In genetics, one method for identifying dominant traits is to pair a specimen with a known hybrid.

Which is the only possible candidate for a stationary distribution?

The only possible candidate for a stationary distribution is the final eigenvector, as all others include negative values. ). Find a stationary distribution for the 2-state Markov chain with stationary transition probabilities given by the following graph: