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Probability II
MAP6473 4810
Homework1b
Prof. JLF King
4May2010
Reading.
Please get your copy of his text as
soon as possible.
Please read Sect.7, pages 92–108 by Monday,
19Jan.
Please read part of Sect.8, pages 111–121 by
Friday, 23Jan.
1b1:
Fix a posint
D
. Let
P
=
P
D

1
be the simplex
of probability vectors
v
∈
R
D
. Fix a
D
×
D
(column)
stochastic
matrix
M
; each column is a prob.vec.
Given a vector
v
∈
P
, deﬁne the
Ces`aro average
v
N
:=
A
N
(
v
) :=
1
N
X
j
∈
[
0
..N
)
M
j
v
.
Prove that
lim
N
→∞
A
N
(
v
)
exists, and is in
P
. [
Hint:
See EE1 on Markov pamphlet.]
1b2:
A
D
×
D
Markov matrix
M
determines transition
probabilities. Use
τ
(
AB
) to denote the transition
prob from state
A
to
B
(it is the
A,B
entry in
M
). A
distribution
σ
() on the states (i.e, an
M
leftinvariant
colvector) determines a onesided Markov process
~
Y
=
Y
0
Y
1
Y
2
...
where the prob of
Y
0
Y
1
...
Y
N
equaling
word
w
0
w
1
...w
N
is the product
σ
(
w
0
)
τ
(
w
0
w
1
)
τ
(
w
1
w
2
)
···
τ
(
w
N

1
w
N
)
.
†
:
Invariance.
Now suppose that
σ
() is an invariant
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This note was uploaded on 01/26/2012 for the course MAP 6473 taught by Professor King during the Fall '11 term at University of Florida.
 Fall '11
 King

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