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
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 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 +
Bibliography
[1] Ahlfors, Lars V. Complex analysis. An introduction to the theory of
analytic functions of one complex variable. Third edition. International
Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New
York, 1978. xi+331 pp.
[2] Dym,
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 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 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