Is the following statement always true? “If an irreducible Markov chain has period 2, then for...

Is the following statement always true? “If an irreducible Markov chain has period 2, then for every state i∈S we have P2ii>0.” (Prove if “yes”, provide a counterexample if “no”)

Expert Solution

The period of a state is defined by

In our case, we have

Consider the Markov chain with the following transition matrix

Its transition diagram is the following:

Notice that all states are strongly connected to each other, making the chain irreducible. Moreover, all state has period . This is obvious for all states other than . For state , period is

Notice that because there is no length- path in the transition diagram, from state to itself. This counterexample shows that the statement in question is not true.

Colby Messinger answered 2 years ago

Stochastic Processes: 1. What does it mean for a Markov Chain to be irreducible? 2. What...

Stochastic Processes: 1. What does it mean for a Markov Chain to be irreducible? 2. What simple conditions imply that a Markov Chain is irreducible?

For an irreducible Markov chain, either all states are positive recurrent or none are. Prove.

Make a list of examples of Markov chains with different properties : irreducible, regular, has a...

Make a list of examples of Markov chains with different properties : irreducible, regular, has a limiting distribution, does not have a limiting distribution, does not have a limiting matrix, has infinitely many stationary distributions.

The following is the transition probability matrix of a Markov chain with states 1, 2, 3,...

The following is the transition probability matrix of a Markov chain with states 1, 2, 3, 4 P 0 1 2 3 0 .4 .3 .2 .1 1 .2 .2 .2 .4 2 .25 .25 .5 0 3 .2 .1 .4 .3 If Xnot = 1 (a) find the probability that state 3 is entered before state 4; (b) find the mean number of transitions until either state 3 or state 4 is entered.

Consider the following Markov chain with P{X0 = 2} = 0.6 and P{X0 = 4} =...

Consider the following Markov chain with P{X0 = 2} = 0.6 and P{X0 = 4} = 0.4: 1 2 3 4 5 6 1 0 0 0 0 1 0 2 .2 .05 0 .6 0 .15 3 0 0 .8 0 0 .2 4 0 .6 0 .2 0 .2 5 1 0 0 0 0 0 6 0 0 .7 0 0 .3 a. What is P{X1 = 4, X2 = 6 | X0 = 2}? b. What...

For each of the following statements, determine if the statement is always, sometimes, or never true....

For each of the following statements, determine if the statement is always, sometimes, or never true. Justify your statement with a proof. Hint: To prove that something isn’t always true, it is sufficient to provide a counterexample. (a) If L is an unrecognizable language, then L is (always/sometimes/never) undecidable. (b) If L is a recognizable language then L COMPLEMENT is (always/sometimes/never) recognizable.

The following is the transition probability matrix of a Markov chain with states 1,2,3,4 ⎛⎞ .4...

The following is the transition probability matrix of a Markov chain with states 1,2,3,4 ⎛⎞ .4 .3 .2 .1 P=⎜.2 .2 .2 .4⎟ ⎝ .25 .25 .5 0 ⎠ .2 .1 .4 .3 (a) find the probability that state 3 is entered before state 4; If X0 = 1 (b) find the mean number of transitions until either state 3 or state 4 is entered.

Statement evaluation. Indicate whether each of the following statements is (i) always true, (ii) sometimes true,...

Statement evaluation. Indicate whether each of the following statements is (i) always true, (ii) sometimes true, or (iii) never true. For those that are (ii) sometimes true; explain when the statement is true. a. All necessary XBRL tags were developed by the XBRL Consortium. b. Companies must use software to prepare XBRL instance documents. c. Companies that do business in a single country do not need XBRL. d. DiversifiedorganizationsshoulduseXBRL. e. General ledger software can create XBRL tags. f. OrganizationsthatadoptXBRLmustcreatetheirownnamespace. g....

Which of the following statement is the most true: A) In period of rising pricing FIFO...

Which of the following statement is the most true: A) In period of rising pricing FIFO tend to give higher value for inventory & higher reported CFO than LIFO. B) In period of rising pricing LIFO tend to give higher value for CGS& lower income & higher reported assets in the balance sheet comparing to FIFO. C) Comparing to FIFO, LIFO provide better matching of expenses with revenue only during a period of rising prices. D) LIFO always provide more...

State whether each of the following is always true (T) or not always true (F). a)...

State whether each of the following is always true (T) or not always true (F). a) If X is a random variable, Corr X, (1/3)X= (1/3). b) If X and Y are independent random variables then E(X|Y ) = E(X) c) d) If fx(x) is the marginal density of a random variable X and fy(y|X = x) is the conditional density of a random variable Y , given a particular realization x of X, then the joint density of X...

Question