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What does it mean for a Markov chain to be recurrent?
A recurrent state has the property that a Markov chain starting at this state returns to this state infinitely often, with probability 1. A transient state has the property that a Markov chain starting at this state returns to this state only finitely often, with probability 1.
What does it mean for a state to be recurrent?
In general, a state is said to be recurrent if, any time that we leave that state, we will return to that state in the future with probability one. On the other hand, if the probability of returning is less than one, the state is called transient.
Is there a Markov chain with no recurrent States?
If all states in an irreducible Markov chain are null recurrent, then we say that the Markov chain is null recurrent. If all states in an irreducible Markov chain are transient, then we say that the Markov chain is transient.
How can you tell if a Markov chain is recurrent?
Let (Xn)n>o be a Markov chain with transition matrix P. We say that a state i is recurrent if Pi(Xn = i for infinitely many n) = 1. Pi(Xn = i for infinitely many n) = 0. Thus a recurrent state is one to which you keep coming back and a transient state is one which you eventually leave for ever.
What is the period of a transient state?
A system is said to be transient or in a transient state when a process variable or variables have been changed and the system has not yet reached a steady state. The time taken for the circuit to change from one steady state to another steady state is called the transient time.
What is steady state in Markov chain?
A well-known theorem of Markov chains states that the probability of the system being in state j after k time periods, given that the system begins in state i, is the (i, j) entry of . The vector containing these long-term probabilities, denoted , is called the steady-state vector of the Markov chain.
Is stationary distribution unique?
Assuming irreducibility, the stationary distribution is always unique if it exists, and its existence can be implied by positive recurrence of all states. The stationary distribution has the interpretation of the limiting distribution when the chain is ergodic.
When is a Markov chain null recurrent?
In a Markov chain with finitely many states, all recurrent states are positive-recurrent. So in your example, the state { 3 } is positive-recurrent. A null-recurrent state is a state the Markov chain will almost-surely return to, but the average of the return time will be infinite.
How does a Markov chain work?
A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The defining characteristic of a Markov chain is that no matter how the process arrived at its present state, the possible future states are fixed.
What is Markov chain applications?
It is named after the Russian mathematician Andrey Markov . Markov chains have many applications as statistical models of real-world processes , such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics.
What is a Markov chain?
Russian mathematician Andrey Markov . A Markov chain is a stochastic process with the Markov property. The term “Markov chain” refers to the sequence of random variables such a process moves through, with the Markov property defining serial dependence only between adjacent periods (as in a “chain”).