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. Liapounov's 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.

roposition

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

1. ,

2. ,

3. .

Then for the following two conditions A,B to hold

A. converges in distribution to ,

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

it is necessary and sufficient that for each we have

 (Lindeberg-Feller condition)

Proof

(Sufficiency). We follow the proof of the proposition ( Liapounov CLT ). As before, we show that because then the statement would follow from the proposition ( Convergence of pm and chf 2 ).

To prove that we verify 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: for some fixed and numbers , . Hence, Using the condition 3 of the proposition we rewrite the above as The first term tends to zero according to the condition According to the condition we have hence We use the proposition ( Liapounov inequality ) for , and to conclude or This shows that because the can be arbitrarily small. The verification of the rest is similar.

 Notation. Index. Contents.