Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
A. Lyapunov central limit theorem.
B. Lindeberg-Feller central limit theorem.
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.

Lindeberg-Feller central limit theorem.


(Lindeberg-Feller CLT). Let MATH MATH is a family of r.v. such that

1. MATH ,

2. MATH ,

3. MATH .

Then for the following two conditions A,B to hold

A. MATH converges in distribution to MATH ,

B. MATH is holospoudic (see the definition ( Holospoudic ))

it is necessary and sufficient that for each $\eta>0$ we have

MATH (Lindeberg-Feller condition)


(Sufficiency). We follow the proof of the proposition ( Lyapunov CLT ). As before, we show that MATH because then the statement would follow from the proposition ( Convergence of p.m. and ch.f. 2 ).

To prove that MATH we verify conditions of the proposition ( Convergence lemma for family of complex numbers ) pointwise in $t$ for MATH : MATH We substitute the Taylor decomposition of $e^{itx}$ around $t=0$ in the following form: MATH for some fixed $\eta$ and numbers MATH , MATH . Hence, MATH Using the condition 3 of the proposition we rewrite the above as MATH The first term tends to zero according to the condition MATH According to the condition MATH we have MATH hence MATH We use the proposition ( Lyapunov inequality ) MATH for $r=2$ , $s=3$ and MATH to conclude MATH or MATH This shows that MATH because the $\eta$ can be arbitrarily small. The verification of the rest is similar.

Notation. Index. Contents.

Copyright 2007