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Unformatted text preview: An idiosyncratic introduction to stochastic processes Class notes for Math 216 Notes  Fall 2010 Jonathan C. Mattingly September 15, 2010 1 Finite State Markov Chains A discrete time stochastic process ( X n ) n ≥ is a collection of random variables indexed by the nonnegative integers Z + = { n ∈ Z : n ≥ } . The set in which the X n take values is called the state space of the stochastic process. Definition. A stochastic process ( X n ) n ≥ is a Markov chain if P ( X n +1 = j  X n = i n , ··· ,X = i ) = P ( X n +1 = j  X n = i n ) for all j,i n , ··· ,i ∈ I . Definition. A Markov chain is time homogeneous if for all k ∈ Z + and i,j ∈ I P ( X k +1 = i  X k = j ) = P ( X 1 = i  X = j ) Unless we say otherwise we will always assume that all Markov chains are time homogeneous. In such cases we will write p n ( i,j ) = P ( X n = j  X = i ) By the Markov property one has P ( X n = x n ,X n 1 x n 1 , ··· X 1 = x 1  X = x ) = p 1 ( x n ,x n 1 ) p 1 ( x n 1 ,x n 2 ) ··· p 1 ( x 1 ,x ) We will begin by concentrating on stochastic processes on a finite state space I . With out loss of generality, we can take the state space to be I = { , 1 ....,N } . 1.1 Markov chains and matrices There is a very fruitful correspondence between finite state Markov chains and Matrices. We begin by considering random variables on a state space I = { ,...,N 1 } . Such a random variable X can be specified completely by N nonnegative numbers { λ i : i ∈ I } such that P ( X = i ) = λ i . Clearly we have that ∑ i ∈ I λ i = 1. It is convenient to organize the λ i in a rowvector λ = ( λ ,...,λ N 1 ) ∈ R N . The vector λ is called the distribution of the random variable X . With this in mind we make the following definition. Definition. A row vector λ = ( λ ,...,λ N 1 ) ∈ R N called a distribution if λ i ≥ 0. If in addition ∑ N 1 i =0 λ i = 1, it is called a probability distribution . 1 Let P ∈ R N,N be a matrix with nonnegative entries. We will write P i,j for the i jth entry of P , that is to say P = p , ··· p ,N 1 . . . . . . . . . p N 1 , ··· p N 1 ,N 1 Definition. A square matrix P with nonnegative entries is called a stochastic matrix if all rows sum to one. That is to say, for all j , ∑ i P ji = 1. Stochastic matrices are in onetoone correspondence with time homogeneous Markov processes on a finite state space. The correspondence is given by P ij = P ( X 1 = j  X = i ) It then follows that P ( X n = j  X = i ) = ( P n ) ij Or in other words, the distribution of the random variable X n when conditioned to have X = i is given by the row vector ( P n ) i, * by which we mean the i th row of the matrix P n . In other words, ( ( P n ) i, ,..., ( P n ) i,N 1 ) (1) If we denote by e ( i ) the row vector with 1 in the i th slot and 0 in the remaining slots, then (1) can be written compactly as e ( i ) P n ....
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This note was uploaded on 01/16/2011 for the course MATH 216 taught by Professor Mckinley,s during the Fall '08 term at Duke.
 Fall '08
 Mckinley,S
 Markov Chains

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