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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.
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.

Lyapunov central limit theorem.


roposition

(Convergence lemma for family of complex numbers) Let MATH MATH be a family of complex numbers such that

1. MATH , $n\rightarrow\infty$ ,

2. MATH for a constant $M$ independent of $n$ ,

3. MATH .

Then MATH

Proof

We calculate MATH All the potential difficulties within the above calculation are resolved by observation that the $\theta_{nj}$ are small starting from some $n$ according to the condition 1.

Proposition

(Lyapunov CLT) Let MATH MATH is a family of r.v. such that MATH Then MATH

Proof

It suffices to establish 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 the 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 Hence MATH Since MATH we have MATH Then by the proposition ( Lyapunov inequality ), MATH Hence, MATH The condition MATH is verified in a similar manner with the use of MATH .

Finally, MATH Hence, MATH and the proposition ( Convergence lemma for family of complex numbers ) yields MATH





Notation. Index. Contents.


















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