Is (xn)n0 a Markov chain?
, Xn=in) =P(Xn+1=in+1 |Xn=in) =pinin+1.For short, we say (Xn)n?0 is Markov(?, P). Checking conditions (i) and (ii) isusually the most helpful way to determine whether or not a given random process(Xn)n?0 is a Markov chain. However, it can also be helpful to have the alternativedescription which is provided by the following theorem.
What is astationary distribution in a Markov chain?
Suppose a distribution?onSis such that, if our Markov chain starts out with initialdistribution?0=?, then we also have?1=?. That is, if the distribution at time 0 is?,then the distribution at time 1 is still ?. Then?is called astationary distributionforthe Markov chain.
What do you need to know about Markov chain theory?
understand the notion of a discrete-time Markov chain and be familiar with boththe ?nite state-space case and some simple in?nite state-space cases, such asrandom walks and birth-and-death chains; know how to compute for simple examples the n-step transition probabilities,hitting probabilities, expected hitting times and invariant distribution;
What is the limiting fraction of time a Markov chain spends?
(1.39) Theorem. Let X0, X1, . . . be a Markov chain starting in the stateX0 =i, andsuppose that the statei communicates with another statej. The limiting fraction of timethat the chain spends in statej is11/EjTj. That is, 11lim 0 =n??nXI{Xt=j}=t=1EjTjwith probability 1. So we assume thatjis recurrent.