Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.

Convergence in Lp.


efinition

The random variable belongs to the class MATH iff MATH . The quantity MATH is called the $L^{p}-$ norm. The sequence MATH converges in $L^{p}$ to $X\in L^{p}$ iff MATH .

Proposition

(Convergence in Lp and in probability 1) If $X_{n}\rightarrow X$ in $L^{p}$ then $X_{n}\rightarrow X$ in probability.

Proof

By the formula ( Chebyshev inequality ) with MATH we have MATH and the claim follows.

Proposition

(Convergence in Lp and in probability 2) If $X_{n}\rightarrow X$ in probability and $\exists Y\in L^{p}$ s.t. MATH for all $n$ then $X_{n}\rightarrow X$ in $L^{p}$ .

Proof

Fix a small $\varepsilon>0$ , then MATH The last expectation converges to 0 because MATH converges to 0.





Notation. Index. Contents.


















Copyright 2007