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What does the keyword discrete time refer to in discrete time Markov chain?
A discrete-time Markov chain is a sequence of random variables. with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states: if both conditional probabilities are well defined, that is, if.
Is Markov chain discrete?
A Markov chain is a Markov process with discrete time and discrete state space. So, a Markov chain is a discrete sequence of states, each drawn from a discrete state space (finite or not), and that follows the Markov property.
How do you specify a Markov chain?
A Markov chain is irreducible if there is one communicating class, the state space. is finite and null recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties—that is, if one state has the property then all states in its communicating class have the property.
What happens when a Markov chain is aperiodic?
Consequently, if the Markov chain is irreducible, then all states have the same period. The proof is another easy exercise. There is a simple test to check whether an irreducible Markov chain is aperiodic: If there is a state i for which the 1 step transition probability p(i,i)>. 0, then the chain is aperiodic.
Which is the most common Markov chain in scalar multiplication?
10 MARKOV CHAINS: BASIC THEORY. scalar multiplication by positive integers, that is, if m 2Ax then all multiples of m are elements of Ax . Second, the greatest common divisor of the numbers in Ax is 1, because by hypothesis the Markov chain is aperiodic.
Which is the transition probability matrix of the chain?
Define the transition probability matrix P of the chain to be the XX matrix with entries p(i,j ), that is, the matrix whose ith row consists of the transition probabilities p(i,j )for. j 2X: (4) P=(p(i,j ))i,j 2X. If Xhas N elements, then P is an N N matrix, and if Xis infinite, then P is an infinite by infinite matrix.