since P(X1 = · X0 = i) is a probability distribution function Theorem Let λ be a distribution (on S) and P a stochastic matrix The sequence X = (X0,X1,
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[PDF] 9 Markov Chains: Introduction
Markov Chains: A discrete-time stochastic process X is said to be a Markov Chain if it has the Markov Property: Markov Property (version 1): For any s, i0, ,in−1 ∈ S and any n ≥ 1, P(Xn = sX0 = i0, ,Xn−1 = in−1) = P(Xn = sXn−1 = in−1)
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The process X is a Markov chain if it satisfies the Markov property: P(Xn+1 Show that every transition matrix on a finite state space has at least one closed
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[PDF] Part IB - Markov Chains (Theorems with proof) - Dexter Chua
since P(X1 = · X0 = i) is a probability distribution function Theorem Let λ be a distribution (on S) and P a stochastic matrix The sequence X = (X0,X1,
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that X◦,X1,X2 ททท is a Markov chain with state space Z/n = {0,1,2,ททท ,n − 1} Show that the sequence of random variables Y◦,Y1,Y2,ททท where Yj = Sj
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Part IB - Markov Chains
Theorems with proof
Based on lectures by G. R. Grimmett
Notes taken by Dexter Chua
Michaelmas 2015
These notes are not endorsed by the lecturers, and I have modiified them (oftensigniificantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.Discrete-time chains
Deifinition and basic properties, the transition matrix. Calculation ofn-step transition probabilities. Communicating classes, closed classes, absorption, irreducibility. Calcu- lation of hitting probabilities and mean hitting times; survival probability for birth and death chains. Stopping times and statement of the strong Markov property. [5] Recurrence and transience; equivalence of transience and summability ofn-step transi- tion probabilities; equivalence of recurrence and certainty of return. Recurrence as a class property, relation with closed classes. Simple random walks in dimensions one, two and three. [3] Invariant distributions, statement of existence and uniqueness up to constant multiples. Mean return time, positive recurrence; equivalence of positive recurrence and the existence of an invariant distribution. Convergence to equilibrium for irreducible, positive recurrent, aperiodic chains *and proof by coupling*. Long-run proportion of time spent in a given state. [3] Time reversal, detailed balance, reversibility, random walk on a graph. [1] 1 ContentsIB Markov Chains (Theorems with proof)Contents0 Introduction 3
1 Markov chains 4
1.1 The Markov property . . . . . . . . . . . . . . . . . . . . . . . . .
41.2 Transition probability . . . . . . . . . . . . . . . . . . . . . . . .
52 Classiification of chains and states 6
2.1 Communicating classes . . . . . . . . . . . . . . . . . . . . . . . .
62.2 Recurrence or transience . . . . . . . . . . . . . . . . . . . . . . .
62.3 Hitting probabilities . . . . . . . . . . . . . . . . . . . . . . . . .
112.4 The strong Markov property and applications . . . . . . . . . . .
132.5 Further classiification of states . . . . . . . . . . . . . . . . . . . .
133 Long-run behaviour 15
3.1 Invariant distributions . . . . . . . . . . . . . . . . . . . . . . . .
153.2 Convergence to equilibrium . . . . . . . . . . . . . . . . . . . . .
194 Time reversal 22
20 IntroductionIB Markov Chains (Theorems with proof)0 Introduction
31 Markov chainsIB Markov Chains (Theorems with proof)1 Markov chains
1.1 The Markov property
Proposition.
(i)λis adistribution, i.e.λi≥0,P iλi= 1. (ii)Pis astochastic matrix, i.e.pi,j≥0 andP jpi,j= 1 for alli.Proof.
(i)Ob vioussince λis a probability distribution.
(ii)pi,j≥0 sincepijis a probability. We also have X jp i,j=X jP(X1=j|X0=i) = 1sinceP(X1=· |X0=i) is a probability distribution function.Theorem.Letλbe a distribution (onS) andPa stochastic matrix. The
sequenceX= (X0,X1,···) is a Markov chain with initial distributionλand transition matrixPifff P(X0=i0,X1=i1,···,Xn=in) =λi0pi0,i1pi1,i2···pin-1,in(∗) for alln,i0,···,in Proof.LetAkbe the eventXk=ik. Then we can write (∗) as P(A0∩A1∩ ··· ∩An) =λi0pi0,i1pi1,i2···pin-1,in.(∗) We ifirst assume thatXis a Markov chain. We prove (∗) by induction onn. Whenn= 0, (∗) saysP(A0) =λi0. This is true by deifinition ofλ.Assume that it is true for alln < N. Then
P(A0∩A1∩ ··· ∩AN) =P(A0,···,AN-1)P(A0,···,AN|A0,···,AN-1)
So it is true forNas well. Hence we are done by induction. Conversely, suppose that (∗) holds. Then forn= 0, we haveP(X0=i0) =λi0.