STAT333 Lecture Notes - Lecture 9: Equivalence Class, Identity Matrix, Random Variable

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4 Feb 2016
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Discrete-time Markov Chains
Generally speaking, {X(t), t T}is called a stochastic process if X(t) is a random
variable (or random vector) for any fixed tT.Tis referred to as the “index set”, and
is often interpreted in the context of time. As such, X(t) is often called the “state of the
process at time t”.
The index set Tcan be a continuum of values (e.g., T={t:t0}) or can be a set of
discrete points such as T={t0, t1, t2, . . .}. Since there is a one-to-one correspondence with
the set of non-negative integers {0,1,2, . . .}, we will use such an index set for a discrete-time
process. In other words, {X(n), n = 0,1,2, . . .}(or {Xn, n = 0,1,2, . . .}) will represent a
general discrete-time stochastic process.
Definition: A stochastic process {Xn, n = 0,1,2, . . .}is said to be a discrete-time Markov
chain (MC) if the following conditions hold true:
(i) For any n= 0,1,2, . . . ,Xntakes on a finite or countably infinite set of possible
values (unless otherwise stated, we assume that the general state space for Xnis
{0,1,2, . . .}).
(ii) For any n= 0,1,2, . . . ,
P(Xn+1 =xn+1|Xn=xn, Xn1=xn1, . . . , X0=x0) = P(Xn+1 =xn+1|Xn=xn).
The above relation implies that the conditional distribution of any future state Xn+1
given the past states X0, X1, . . . , Xn1and the present state Xnis independent of the
past states. We refer to this relation as the Markov property.
Remark: In general, the sequence of random variables {Xn}
n=0 are neither independent
nor identically distributed.
Definition: For any pair of states iand j, the transition probability from state iat time n
to state jat time n+ 1 is given by Pn,i,j =P(Xn+1 =j|Xn=i) for n= 0,1,2, . . . .
Let Pn= [Pn,i,j ] be the associated matrix containing the elements Pn,i,j , referred to as the
one-step transition probability matrix (TPM) from time nto time n+ 1. It looks like:
Pn=
Pn,0,0Pn,0,1Pn,0,2· · · Pn,0,j · · ·
Pn,1,0Pn,1,1Pn,1,2· · · Pn,1,j · · ·
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Pn,i,0Pn,i,1Pn,i,2· · · Pn,i,j · · ·
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