Department

MathematicsCourse Code

MATH 303Professor

Marco FerranteStudy Guide

QuizThis

**preview**shows half of the first page. to view the full**2 pages of the document.**MATH 303 – Assigment 2: Due Wednesday, January 23 at start of class

I. Problems to be handed in:

1. A Markov chain {Xn, n ≥0}with state space S={0,1,2,3,4,5}has transition proba-

bility matrix

P=

α0 1 −α0 0 0

β/2 0 0 1/3 2/3−β β/2

0 0 1 0 0 0

0β/2β/2 1/2 0 0

0 0 0 1 −γ0γ

γ1−γ0 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 the states in each class are recurrent or transient and ﬁnd

their period (or determine that they are aperiodic).

2. An equivalent class C of communicating states, called a communicating class, is said to

be closed if Pi,j = 0 whenever i∈Cand j6∈ 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 com-

municating class.

(c) Find an example of a Markov chain with no closed communicating class.

3. Divide the circle into Npoints, and consider a particle randomly moving between these

points, in such a way that the position of the particle is a Markov Chain, and at each

step the particle jumps to a neighboring point with equal probabilities:

1

2

1

2

0

1

2

···

N−1

Compute P(Xn= 0|X0= 0) for all n≥0. Hint: Use the discrete Fourier transform to

diagonalize the transition matrix. That is, express the transition matrix in the orthonor-

mal basis ˆe(0),ˆe(1),...,ˆe(N−1) of CNwhich has the following components in the standard

basis: ˆe(k)

j=1

√Ne2πi

Njk, (j= 0, . . . , N −1, i=√−1).

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