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 be a family of complex numbers such that

1. , ,

2. for a constant independent of ,

3. .

Then

Proof

We calculate All the potential difficulties within the above calculation are resolved by observation that the are small starting from some according to the condition 1.

Proposition

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

Proof

It suffices to establish that because then the statement would follow from the proposition ( Convergence of p.m. and ch.f. 2 ).

To prove that we verify the conditions of the proposition ( Convergence lemma for family of complex numbers ) pointwise in for : We substitute the Taylor decomposition of around in the following form: Hence Since we have Then by the proposition ( Lyapunov inequality ), Hence, The condition is verified in a similar manner with the use of .

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

 Notation. Index. Contents.