An equivalent class C of communicating states, called a communicating class, is said to be closed if Pi,j = 0 whenever i ∈ C and j 6∈ C. In other words, a communicating class is closed if there is no escape from that class.
(a) Show that every recurrent class is closed.
(b) Show that every Markov chain on a finite state space has at least one closed communicating class.
(c) Find an example of a Markov chain with no closed communicating class.
An equivalent class C of communicating states, called a communicating class, is said to be closed...
A Markov chain is said to be a tree process if 75. (i) Pii 0 whenever Pi > 0, (ii) for every pair of states i and , i夭, there is a unique sequence of distinct states l = 10, 11, . . . , In-1, In-1 such that k=0,1 0, > 4,4+1 In other words, a Markov chain is a tree process if for every pair of distinct states i and j there is a unique way for the...
2. A Markov Chain with a finite number of states is said to be regular if there exists a non negative integer n such that for any i, J E S, Fini > 0 for any n-มิ. (a) Prove that a regular Markov Chain is irreducible. (b) Prove that a regular Markov Chain is aperiodic (c) Prove that if a Markov Chain is irreducible and there exists k E S such that Pk0 then it is regular (d) Find an...
2. A Markov Chain with a finite number of states is said to be regular if there exists a non negative integer n such that for any i, j E S, > 0 for any n 兀 (a) Prove that a regular Markov Chain is irreducible. (b) Prove that a regular Markov Chain is aperiodic. (c) Prove that if a Markov Chain is irreducible and there exists k e S such that Pk>0 then it is regular (d) Find an...
-1,2,3,4,5,63 and transition matrix Consider a discrete time Markov chain with state space S 0.8 0 0 0.2 0 0 0 0.5 00 0.50 0 0 0.3 0.4 0.2 0.1 0.1 0 0 0.9 0 0 0 0.2 0 0 0.8 0 0.1 0 0.4 0 0 0.5 (a) Draw the transition probability graph associated to this Markov chain. (b) It is known that 1 is a recurrent state. Identify all other recurrent states. (c) How many recurrence classes are...
Consider the Markov chain with state space S = {0,1,2,...} and transition probabilities I p, j=i+1 pſi,j) = { q, j=0 10, otherwise where p,q> 0 and p+q = 1.1 This example was discussed in class a few lectures ago; it counts the lengths of runs of heads in a sequence of independent coin tosses. 1) Show that the chain is irreducible.2 2) Find P.(To =n) for n=1,2,...3 What is the name of this distribution? 3) Is the chain recurrent?...
Let Xo, X1, denote a Markov chain on the nonnegative integers with transition prob- abilities po,j aj, j > 0, where aj > 0 and Σ000 aj 1; and for i > 1, pi,i r and Pii-1-1-r with r E [0, 1]. Let M = sup{] > 0 : ai > 0}. Hint: Drawing the state diagram will be helpful.] (a) For Y = 1 and a0 1, find all the recurrent classes if there is any. (b) For 0
My Professor of Stochastic Processes gave us this
challenge to be able to exempt the subject, but I cant solve
it.
Stochastic Processes TOPICS: Asymptotic Properties of Markov Chains May 25, 2019 1.Consider the stochastic process R-fRnh defined as follows: Where {Ynjn is a succession of random variable i.i.d (Independent random variables and identically distributed), with values in {1,2, ...^ with Ro 0 a) Why R is a Markov Chain? Find the state space of R b) Find the transition...
Exercise 5.10. Let P be the transition matrix of a Markov chain (Xt)120 on a finite state space Ω. Show that the following statements are equivalent: (i) P is irreducible and aperiodic (ii) There exists an integer r 0 such that for all i,je Ω, (88) (ii) There exists an integer r 20 such that every entry of Pr is positive.
A Markov chain {Xn, n ≥ 0} with state space S = {0, 1, 2, 3, 4,
5} has transition probability matrix P.
ain {x. " 0) with state spare S-(0 i 2.3.45) I as transition proba- bility matrix 01-α 0 0 1/32/3-3 β/2 0 β/2 0 β/2 β/21/2 0001-γ 0 0 0 0 (a) Determine the equivalence classes of communicating states for any possible choice of the three parameters α, β and γ; (b) In all cases, determine if...
Determine the equivalence classes and classify the states as
transient or recurrent for a Markov chain with the following
transition matrices c and d. Also determine the closed and
irreducible subsets of the state space.
0 10.4 1 1 0 0.1 0.7 0.4 c) 0.1 0.3 10.3 0 0 0.3 0 0.2 0 0 0.5 0 0 0 0 0 0 d) 0 0.5 0.3 1 0 0.7 0.1 0.2 0 0 0 0 0.5 0 0 0 0 0...