6.262 Lect. 6 2/19/2010
Discrete Stochastic Processes
1
DISCRETE
STOCHASTIC
PROCESSES
Lecture 6
Convergence of Sequences of Random Variables
Review:
Sequences of random variables: four examples
Four progressively weaker types of convergence
(Review):
Sure convergence of a sequence of random variables.
Almost sure convergence and the strong law of large numbers
Convergence in probability and the weak law of large numbers.
(Review):
Convergence in distribution and the central limit theorem.
(Review):
Convergence in mean-square
The Zero - One Laws of Borel and Cantelli
Axiom: Probability is Countably Additive
Lemma: Probability has a Continuity-Like Property
Application: Continuity of CDF from Right but not from Left
Statement of Zero-One Law of Borel and Cantelli
Proof of Borel-Cantelli Lemma
Examples and Applications

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
6.262 Lect. 6 2/19/2010
Discrete Stochastic Processes
2
Math Notation
∀
means “for each” (equivalently, “for every” or “for all”).
∃
means “there exists”.
A
B
⇒
means that “whenever
A
is true,
B
is also true.”
s.t. means “such that”.
n
X
X
→
means “
n
X
converges to
X
as
n
→∞
,”
though we also need to specify
what sort of objects
n
X
and
X
are (numbers, functions, random variables) and what
kind of convergence we mean.