### 2. Stationary DistributionsRecall that a stationary distribution of a Markov chain is a probability distribution \( P(X) \) such that starting with it and applying the transition probabilities does not change it. That is, it satisfies \[ P(X = x) = \sum_{x'} P(X_{t+1} = x \, | \, X_t = x') P(X = x'). \]#### (a)Give an example of a Markov chain with two states and deterministic transitions where applying the mini forward algorithm does not necessarily converge to a stationary distribution.#### (b)Show, nevertheless, that your example has a stationary distribution.

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Answered: 2. Stationary Distributions Recall that…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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4. Let (X₂)o be a Markov chain with state space S = {1,2,3} and transition matrix .8 .2 O .5 .5 0 0 0 1 Find two different stationary distributions for this Markov chain. P =
A Markov chain {s} with state space n={1, 2, 3} has a sequence of realizations and process transitions as follows: Transition (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3.1) (3,2) (3,3) t S: S+1 2 1 1. 1 1 1 1 1 3 1 3 3 3 4 3 1. 2 3 1. 2 1 7 2 2 1. 8 2 2 1 9 2 3 1. 10 3 11 1 1 12 1 1. 13 1 1. 14 1. 1. 15 3 1. 1. 16 1. Number of frequencies : 1 3. 2 2 1 1 1 1 a. determine the transition frequency matrix O= 2 12 93 021 022 °23 %3D 31 °32 °33) 1 P1 P12 P13 b. determine the estimated transition probability matrixP= 2 P21 P22 P23 3 P31 P32 P33)
[stochastic process1]
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
9. Let i be a transient state of a continuous-time Markov chain X with X (0) : = i. Show that the total time spent in state i has an exponential distribution.
Find the limiting distribution for this Markov chain. Then give an interpretation of what the first entry of the distribution you found tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology.
c) Consider the transition Markov chain with three states S = (1, 2, 3) as shown below. %3D 1 Given that; P(X, = 1) = 3' fine P(X, = 1,X, = 2,X2 = 3) N/3 1/2 1/2 1/4 1/2 1/4
8. List the Gauss–Markov conditions required for applying a t & F-tests.
C)
For each of the following transition matrices, do the following: (1) Determine whether the Markov chain is irreducible. (2) Find a stationary distribution 7; is the stationary distribution unique? (3) Give the period of each state. (4) Without using any software package, find P200 approximately. (a) (1/4 1/4 1/2 P = | 1/3 2/3 1/2 1/2, (Ъ) (0.2 0.8 0.5 0.5 P = 0.3 0.7 1
1. A Markov chain has state space {0, 1, 2} and transition probability matrix .1 .5 .4 P = 0 0 1 .2 .3 .5 (a) the probability of going from state 0 to state 2 in 2 transitions (b) the steady-state distribution, if it exists Determine:
A path of length k in a Markov chain {X,,n= 0,1,...} is a sequence of states visited from step n to step n+k. Let pij be the transition probability from state i to state j. Show that starting from state i,, the probability that the chain follows a particular path i, - i„.1 - in2 - ... - i,t is given by %3D "x*\":= "X)d = i2.,Xpsk = inrk | X, = i,) n+1 n+l> n+2 n+k = P Pu.

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