HW 1, due Wednesday April 14

Chapter 13, Problems 6,7,8,12,13,14,23,40,50

The distribution of grades in HW1

41, 45, 47, 47, 47, 48, 49, 49, 50
--------------------------------------------


HW 2, due  Wednesday April 28

Chapter 13, Problem 63,

Chapter 14, Problems 1, 2, 4, 5, 26,

Also:

(7) Prove that if F(x) and F(n,x), n=1,2,...,
 are distribution functions on R and F(n,x) tends
 to F(x) and F(n,x-) tends to F(x-) for all x,
 then F(n,x) tends to F(x) uniformly on R.
(Hint: While considering a convergent sequence
x -> x  consider two cases when infinitely many of x
 n    0                                             n

are to the left or to the right of x  )
                                    0

(8) Let a be a fixed number >0 and f(x) be an even function
 defined as (a-x)/a for x in [0,a] and as zero for x>a.
 Prove that f(x) is positive definite. (Hint: use
 the inversion formula)

(9) Let f(x) be the function from Problem (8) 
 and let g(x) be defined in the same way as f(x)
 but with a replaced by a constant b>a. Prove that,
 for any c in (0,1), the function max(f,cg) is positive
 definite. (Hint: observe max(f,cg)=cg+max(f-cg,))

The distribution of grades in HW2

21, 37, 40, 41, 45, 46, 49, 49, 50
-------------------------------------------


HW 3 due on Wednesday May 12

Chapter 14, Prove Theorem 19, Problems 43, 45, 47, 57;
(Hint to proving Theorem 19:
Let phi(n,u) be a sequence of moment generating functions
which converges to a function phi such that phi(0+)=1.
While checking tightness you observe that for any
                         -b     -x
x>b>0, it holds that  1-e  < 1-e  ,
so that
           -1        -uX
P{uX>1}(1-e  ) < 1-Ee   .  Hence, whenever u>0
is so small that 1-phi(n,u)<epsilon, we have
                   -1
P{X(n)>1/u}<(1-1/e)  epsilon.)

Chapter 15, Problems 2, 3;

Chapter 25, Problems 3, 4 (in 4 one says that a measure  U on Z_+ has
a bounded density iff the sequence U(n) is bounded).


Grades distribution for NW 3

32, 38, 40, 42, 45, 46, 47, 47, 50


_______________________________________________

Total grades distribution after HW1-3

100, 122, 127, 132, 135, 139, 142, 145, 146

_________________________________________________

HW 4 due Wednesday  May 26
 

(1) (see Problem 25-12) For q in (0,1) and p=1-q
let X be a renewal sequence with waiting time distribution

                     n-1
given by P(T(1)=n)=pq   , n=1,2,... . Take an integer k>0 and
define Y(0)=1, Y(n)=X(k+n) if n>0. Prove that
Y is a renewal sequence with the same waiting time distribution.
(Hint: Use the result of the following problem)

(2) For X from the previous problem, prove that
X(n), n=1,2,..., are iid.
(Hint: For a sequence x(n) of 0's and 1's, n=1,2....
define t(1) as inf{n>0: x(n)=1} and, generally,
t(m)=inf{n>t(m-1): x(n)=1} for m>1. Then
for any integer k>0 let m(k)=inf{m>0:t(m)>k}.
Observe that m(k)-1 is the number of 1's in the
sequence x(1),...,x(k). Prove that

P{X(1)=x(1),...,X(k)=x(k)}=P{T(m)-T(m-1)=t(m)-t(m-1)

for m<m(k) and T(m(k))-T(m(k)-1)>k-t(m(k)-1)}

  m(k)-1  k-(m(k)-1)
=p      q            .

You may also use a different method)

(3) Let X be a Markov sequence in Psi, B be a Borel subset
of Psi and tau=inf{n: X(n) in B}. Define
      x
S(x)=E (tau). Prove that TS=S-1 on the complement of B and S=0
on B.


Chapter 26, Problems 5, 6, 21, 31, 34 (Hint to 21:


E{f(X               ) I      |F }=
     min[(n+1),tau]    tau>n   n


I      E{f(X    )|F }  )
 tau>n      n+1    n

(9) (cf. Problem 26-43)
Show that if the number of states is finite and if there is a
number k such that all entries of the kth power of the 
transition matrix are positive, then
 n                 -1
T (x,y) tends to m      as n goes to infinity for all x,y.
                   y