Consider a sequence of random variables X0, X1, X2, X3, . . . which form a...

Consider a sequence of random variables X0, X1, X2, X3, . . . which form a Markov chain.

(a) Define the Markov property for this Markov chain both in words and using a mathematical formula.

(b) When is a Markov chain irreducible?

(c) Give the definition for an ergodic state.

Expert Solution

a) A sequence of random variables is said to be a Markov Chain if Future actions only depends on present action and independent of all other past actions.

mathematically,

Here 1,2,3...n that is suffix of X are called steps and a₀,a_1,a₂....a_n are called states of the markov chain

b) A Markov chain is said to be irreudicible if every state can be reached from every other state.

Let P_ij⁽ⁿ⁾denote the transition from ith state to jth state at nth step.

Then if markov chain is irreducible then P_ij⁽ⁿ⁾ > 0 ; For some n

c) A state of the Markov chain is said to be ergdic if the state is aperiodic and non null persistent.

A aperiodic state is defined as where there is no specifiq period of returning in a particular state. Say at state 1 the process returns at step 1,4,9,11 etc then the state is aperiodic.

A non null persistent state is where the mean recurrence time is finite.

orchestra answered 2 years ago

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