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.
 A. Weak law of large numbers.
 B. Convergence of series of random variables.
 C. Strong law 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.

## Laws of large numbers.

efinition

The r.v. and are called "uncorrelated" if they have finite second moments and . A family of r.v. is "uncorrelated" if , , .

Proposition

(Simple law of large numbers) If the family of r.v. is uncorrelated and the second moments have a common bound then in , in probability and almost surely.

Proof

We introduce the notation To investigate the convergence in , we calculate The are uncorrelated, hence, the second term is zero: Since the second moments have common bound we conclude that Hence, It follows by the proposition ( Convergence in Lp and in probability 1 ) that also

It remains to prove the a.s. convergence. According to the formula ( Chebyshev inequality ) Hence, if we restrict our attention to the subindexing then Therefore, according to the proposition ( Borel-Cantelli lemma, part 1 ), Then, according to the proposition ( IO criteria for AS convergence ), It remains to consider the middle terms for every . We introduce the notation : for . We estimate for every s.t. :

Hence, it suffices to prove that We calculate The are uncorrelated, hence the cross terms vanish: Therefore, according to the formula ( Chebyshev inequality ), It follows, according to the proposition ( Borel-Cantelli lemma, part 1 ), Then, according to the proposition ( IO criteria for AS convergence ),

 A. Weak law of large numbers.
 B. Convergence of series of random variables.
 C. Strong law of large numbers.
 Notation. Index. Contents.