Chapter 2
Weak Convergence
2.1
Characteristic Functions
If is a probability distribution on the line, its characteristic function is
dened by
(t) =
exp[ i t x ] d.
(2.1)
The above denition makes sense. We write the integrand eitx as cos tx +
i sin tx and
Chapter 7
Dynamic Programming and
Filtering.
7.1
Optimal Control.
Optimal control or dynamic programming is a useful and important concept
in the theory of Markov Processes. We have a state space X and a family
of transition probability functions indexed
Chapter 1
Measure Theory
1.1
Introduction.
The evolution of probability theory was based more on intuition rather than
mathematical axioms during its early development. In 1933, A. N. Kolmogorov [4] provided an axiomatic basis for probability theory and i
Chapter 5
Martingales.
5.1
Denitions and properties
The theory of martingales plays a very important ans ueful role in the study
of stochastic processes. A formal denition is given below.
Denition 5.1. Let (, F , P ) be a probability space. A martingale s
Chapter 3
Independent Sums
3.1
Independence and Convolution
One of the central ideas in probabilty is the notion of independence. In
intuitive terms two events are independent if they have no inuence on each
other. The formal denition is
Denition 3.1. Two
Chapter 6
Stationary Stochastic
Processes.
6.1
Ergodic Theorems.
A stationary stochastic process is a collection cfw_n : n Z of random variables with values in some space (X, B) such that the joint distribution of
(n1 , , nk ) is the same as that of (n1 +
Chapter 4
Dependent Random Variables
4.1
Conditioning
One of the key concepts in probability theory is the notion of conditional
probability and conditional expectation. Suppose that we have a probability
space (, F , P ) consisting of a space , a -eld F
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Limit Theorems.
Set 6. Due Nov 21, 2002.
Q1. Let Xn,1 , Xn,2 , . . . , Xn,n be a triangular scheme of mutually independent uniformly
innitesimal random variables. We are interested in the limiting distribution of Yn =
Xn,1 + Xn,2 + + Xn,n . Assume that Yn
Probability, Limit Theorems
Problem set 7. Due Nov 14, 2002
Q1. If X and Y are independent random variables show that for any p > 0
E [|X + Y |p ] < E [|X |p ] < , E [|Y |p ] <
Q2. Let be a probability distribution on the line. Let be the innitely divisi
Probability, Limit Theorems
Problem set 6. Due Oct 31, 2002
Q1. In the law of the iterated logarithm, where
lim sup
lim inf
X1 + X2 + + Xn
=1
2n log log n
X1 + X2 + + Xn
= 1
2n log log n
prove that the set of limit points of
X1 + X2 + + Xn
2n log log n
co
Limit Theorems.
Set 4. Due Oct 24, 2002.
Q1. Let X1 , X2 , . . . , Xn , . . . be a sequence of independent and identically distributed
1
random variables taking the values 1, each with probability 2 . For a > 0, show that the
probability
X1 + + Xn
pn (a)
Probability, Limit Theorems
Problem set 4. Due Oct 17, 2002
Let X1 , X2 , . . . , Xn , . . . be a sequence of independent random variables such that
E [Xi ] = 0 for i = 1, 2, . . . However they are not assumed to have the same distribution. We are interes
Probability, Limit Theorems
Problem set 3. Due Oct 10, 2002
Q1. If f (x) is a bounded lower semicontinuous function on R and n show that
f (x)d(x) lim inf
n
f (x)dn (x)
Hint: Write f (x) = lim fn (x) an increasing limit of bounded continuous functions.
Q2
Probability, Limit Theorems
Problem set 1. Sept 19, 2002
Q1. If is a probability distribution on R such that |x| d(x) < for some
0 < < 1, show that the characteristic function is Hlder continuous of exponent .
o
Q2. Is there some sort of a converse?
1
Probability, Limit Theorems
Problem set 1. Sept 19, 2002
For each probability measure on the Borel subsets of R2 dene the distribution
function F (a, b) by
F (a, b) = (x, y ) : < x a, < y b
Show that F (, ) satises the following conditions:
1. F (a, b) is
Probability/ Limit Theorems
Final Examination
Due before Dec 19
Q1. For each n, cfw_Xn,j ; j = 1, 2 . . . , n are n mutually independent random variables taking
values 0 or 1 with probabilities 1 pn,j and pn,j respectively. i.e
and P [xn,j = 0] = 1 pn,j
P