STAT4528: Probability and Martingale Theory
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
STAT4528: Probability and Martingale Theory
1. (a) (i) Define sets of real numbers as follows. Let An = (一 
, 1] if n is odd, and
An = (一1, 
] if n is even. Find lim supnAn and lim infnAn .
(ii) Show that if An t A of An l A then lim infnAn = lim supnAn = A.
(b) Show that if µ is a finite measure, there cannot be uncountably many disjoint sets
A such that µ(A) > 0.
(c) Let f be a Borel measurable function from R to R and a e R, and define g(x) = f(x + a). Show that
fdλ = hdλ
R R
in the sense that if one integral exists, so does the other, and the two are equal. (Start with indicators.)
2. (a) Let f and g be extended real-valued Borel measurable functions on (Ω , r), and
define
h(ω) = 
where A is a set in r. Show that h is Borel measurable.
(b) Let X1, X2 , ... be an i.i.d sequence such that E(lX1l) < o. Show that
X1X2 + X2X3 + ... + XnXn+1 
E(X1X2)
as n goes to infinity.
3. Let Xn, n = 1, 2, . . ., and Y be random variables defined on a common probability space
(Ω , r, P). We assume that Xn, n ≥ 1 are independent and such that
P (lXnl ≤ C) = 1, and EXn = 0, n = 1, 2, . . .
where C > 0 is a constant.
Let gm = σ (Xn, n ≤ m) and go = σ (Xn, n ≥ 1).
(a) Show that
lim E (Y l gm) = E (Y l go ) , P 一 a.s.
(b) Assuming that Y is gm-measurable for a certain m ≥ 1, and such that ElY l < o,
show that
lim E (XnY) = 0 .
(c) Using (2) or otherwise, show that
n→o
for arbitrary Y such that ElY l < o.
4. Let εn, n ≥ 1 be a sequence of independent and identically distributed random variables, such that Eεn = 0 and Eεn(2) < o. Let xn n ≥ be sequence of real numbers. Consider a linear regression problem
Yn = axn + on, n ≥ 1,
where a is an unknown slope parameter and Yn are observations. It is well known that that the Least Squares Estimator 
n based on the first n observations Y1, . . . , Yn takes
the form n
(a) Identify a martingale (Mn) such that
(M）n ,
where (M）n denotes the predictable quadratic variation of the martingale (Mn). Note, that you need to specify the filtration as well.
(b) Show that 
n converges to a limit P-a.s.
(c) Show that
lim 
n = a, P 一 a.s.
if and only if
o
xn(2) = o .
n=1
(d) Let εn, n ≥ 1 be a sequence of independent and identically distributed random variables, such that Eεn = 0 and Elεnl < o. Is still the statement in (c) correct?
2022-06-05