MTH 664
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MTH 664 - Lectures 4 & 5
Yevgeniy Kovchegov
Oregon State University
MTH 664
Topics:
DeMoivres and Stirlings formulas.
Bernoulli LLN.
Measure and Integral.
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MTH 664
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DeMoivres and Stirlings formulas.
It is easy to check that n! < nn . A bett
MTH 664
Topics:
Martingales.
Filtration. Stopping times.
Probability harmonic functions.
Optional Stopping Theorem.
Martingale Convergence Theorem.
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Conditional expectation.
Consider a probability space
X F.
(, F , P ) and a random variabl
MTH 664
Topics:
Modes of convergence.
Dirac point-mass a(x).
Statistical independence.
Laws of large numbers (LLN).
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Modes of convergence.
Let (, F , P ) be a probability space, and X1 , X2 , . . . , X are
random variables over (, F ).
We
MTH 664
Topics:
Conditional probability. Independence.
Conditional expectation.
Modes of convergence.
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Conditional probability. Independence.
Consider a probability space (, F , P ).
For two events A and B in (, F ) such that P (B) > 0,
P (
MTH 664
Topics:
Moment generating function.
Cherno bound.
The Large Deviations.
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The Central Limit Theorem (CLT).
The Central Limit Theorem (CLT). Suppose X1 , X2 , . . .
are i.i.d. random variables with mean E[Xj ] = < and
variance V ar(Xj
MTH 664
Topics:
Applications.
Weak convergence of distributions.
The Fourier transform.
Plancharel Theorem.
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Laws of Large Numbers (WLLN vs. SLLN).
Let X1 , X2 , . . . be independent identically distributed (i.i.d.)
random variables on a pr
MTH 664
Topics:
Statistical independence.
Laws of large numbers (LLN).
Borel-Cantelli Lemma.
Kolmogorovs Maximal Inequality.
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Modes of convergence.
Let (, F , P ) be a probability space, and X1 , X2 , . . . , X are
random variables over (,
MTH 664
Topics:
The Fourier transform.
Plancharel Theorem.
The Uniqueness Theorem.
The Convergence Theorem.
The Centarl Limit Theorem.
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Weak convergence in distribution.
We say that probability distributions cfw_n n=1,2,. converge weekly
i
MTH 664
Topics:
DeMoivres and Stirlings formulas.
Bernoulli LLN.
Measure and Integral.
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DeMoivres and Stirlings formulas.
It is easy to check that n! < nn . A better asymptotic analysis of
n! comes in the form of Stirlings formula, and its e
Math 664
Homework #4: Solutions
1. Recall that in homework 3 we proved the One-Series Theorem of Kolmogorov (1930):
If X1 , X2 , . . . are independent mean-zero random variables, and if
2
E [Xj ] < ,
j =1
then
Xj converges almost surely.
j =1
Show that if
Math 664
Homework #3: Solutions
1. Recall that events A1 , . . . , An in a probability space (, F , P ) are independent if
P (Ai1 Ai2 . . . Aik ) = P (Ai1 ) P (Ai2 ) . . . P (Aik )
for any subcollection of distinct indices i1 < i2 < . . . < ik .
Also reca
Math 664
Homework #2: Solutions
1. Let = R,
F = cfw_A R : either A or Ac is countable,
and
P (A) =
0
1
if A is countable
if A is uncountable.
Show that (, F , P ) is a probability measure space.
SOLUTION: First we show that F is a -algebra:
(a) F since c
Math 664
Homework #1: Solutions
1. An urn contains n green and m black balls. The balls are withdrawn one at a time
n
until only those of the same color are left. Show that with probability n+m they are
all green.
SOLUTION: The outcome of the experiment w
MTH 664
Topics:
Measure and Integral.
Random variables.
Expectation.
Jensens inequality.
Convergence theorems.
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Measure and Integral.
Denition 1. A collection F of subsets of is a -algebra if
1. F
2. A F implies Ac F
3. If A1 , A2 , A3 , F