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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
A. Operations on sets and logical statements.
B. Fundamental inequalities.
C. Function spaces.
D. Measure theory.
E. Various types of convergence.
a. Uniform convergence and convergence almost surely. Egorov's theorem.
b. Convergence in probability.
c. Infinitely often events. Borel-Cantelli lemma.
d. Integration and convergence.
e. Convergence in Lp.
f. Vague convergence. Convergence in distribution.
F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
G. Lebesgue differentiation theorem.
H. Fubini theorem.
I. Arzela-Ascoli compactness theorem.
J. Partial ordering and maximal principle.
K. Taylor decomposition.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Uniform convergence and convergence almost surely. Egorov's theorem.


he concept of almost sure convergence was defined by the formula ( Almost sure convergence ). Equivalently, we characterize the set $A$ where the random variables $X_{n}$ converges to the random variable $X$ by the relationship

MATH (Almost sure convergence 2)

Definition

The sequence $X_{n}$ converges to $X$ uniformly on a set $B$ if for MATH MATH s.t. MATH for all $\omega\in B$ .

The key difference from the almost sure convergence is the MATH independence from $\omega$ .

We use the notations MATH for uniform convergence. Using the technique of the section ( Operations on sets and logical statements ) we state that the set $B$ satisfies

MATH (Uniform convergence)
for some function $N\left( m\right) $ .

Uniform convergence implies the convergence a.s. The inverse relationship is given by the following Egorov's theorem.

Theorem

(Egorov theorem). Let $X_{n}\rightarrow X$ a.s. on $A$ then for any small $\delta>0$ there exists a set MATH s.t. MATH and MATH on $B_{\delta}$ .

To prove the Egorov's theorem we need the continuity property of the $\sigma $ -additive measure. It serves as a bridge between the convergence concepts defined in terms of set algebra and the topology introduced by the probability measure.

Lemma

(Continuity lemma). If $P$ is a $\sigma$ -additive probability measure and $A_{n}$ is an increasing ( MATH ) infinite countable collection of sets then MATH

Proof

Introduce the collection of sets MATH according to MATH . The MATH are disjoint and MATH , MATH . It remains to note that MATH and MATH , hence, MATH .

Corollary

If MATH is increasing then MATH . If MATH is decreasing then MATH .

We now proceed with the prove of the Egorov's theorem.

Proof

The a.s. convergence on $A$ implies MATH (see the section ( Operations on sets and logical statements )).

Fix some small $\delta>0$ as required by the conclusion of the theorem. Our goal is to construct the set $B_{\delta}$ as in the formula ( Uniform convergence ) with the property MATH .

Pick some positive integer $m^{\ast}$ . We have MATH because restricting from the intersection in all $m$ to $m^{\ast}$ makes the set bigger. Hence, MATH Consequently, by the lemma ( Continuity lemma ) MATH Note that the set collection MATH increases as $N$ increases, hence the set MATH with $N=S$ is the biggest set in the $N$ -union. Therefore, MATH Choose MATH so that MATH and form the set MATH By the formula ( Uniform convergence ) we have MATH on $B_{\delta}$ . Also, using the formula ( Intersection property ) MATH





Notation. Index. Contents.


















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