APM 504 - PS2 Solutions
1.2.2) Suppose that X1 and X2 are random variables defined on the same probability space.
Then, for any x R, the event
[
cfw_X1 + X2 < x =
cfw_X1 < x q cfw_X2 < q
qQ
is measurable because Q is countable and, for each q, the events
MAT1124H, Random Matrix Theory, Balint Virag
Lecture 1: Wigners Semicircle Law
notes by Fan Zhang
January 7th, 2009
We want to consider symmetric Wigner matrices; more precisely, a symmetric Wigner
matrix M has the following properties:
1. M is symmetric;
STAT 205A - Problem Set 02
Walid Krichene (23265217)
September 17, 2013
Let B be the Borel subsets of R. For B B define
(
1 if (0, ) B for some > 0
(B) =
0 if not
(2.1)
Show that is not finitely additive on B
Show hat is finitely additive but not counta
A LEBESGUE MEASURABLE SET THAT IS NOT BOREL
SAM SCHIAVONE
Tuesday, 16 October 2012
1. Outline
(1)
(2)
(3)
(4)
Ternary Expansions
The Cantor Set
The Cantor Ternary Function (a.k.a. The Devils Staircase Function)
Properties of the Cantor Ternary Function
C
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Lecture 17
Fall 2008
11/5/2008
CONVERGENCE OF RANDOM VARIABLES
Contents
1. Denitions
2. Convergence in distribution
3. The hierarchy of convergence concepts
1
DEFINITIONS
1.1
Almost sure convergence
Den
Tuesday, February 27, 2007
4:05 PM
Ito stochastic integral (cont.)
What about the variance of the Ito stochastic integral for
simple functions?
sdenotes022706b Page 1
Properties of Ito stochastic integral J(f) for simple
functions f:
sdenotes022706b Page
Chapter 2
Wigner Matrices and Semicircular Law
A Wigner matrix is a symmetric (or Hermitian in the complex case) random matrix. Wigner matrices play an important role in nuclear physics and
mathematical physics. The reader is referred to Mehta [212] for a
A non-Borel set
Using transfinite induction (which is beyond the scope of this module) one can show that the cardinality
of the collection B of Borel subsets of R is the same as the cardinality of R. Since this is strictly less
than the cardinality of P(R
Chapter 1
Brownian motion
Note: This chapter and the next are adapted from related chapters of the following
two books as
[1] Fima C Klebaner, Introduction to Stochastic Calculus with Applications, Imperial College Press, 1998.
[2] Steven E. Shreve, Stoch