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.

## Convergence of series of random variables.

roposition

(Kolmogorov inequality for series 1) Let are independent r.v. such that Then for any

Proof

We introduce the notations Observe that We calculate The term vanishes as follows Note that is a function of and is a function of . Hence, these are independent: because the second integral is zero: . We continue the calculation of :

Proposition

(Kolmogorov inequality for series 2) Let be a sequence of independent r.v. such that for every . Then for any we have

Proposition

(Kolmogorov's three series theorem) Let be independent r.v. Fix a constant and define Then the series converges a.s. iff all of the following series converge:

1. ,

2. ,

3. .

Proof

Suppose that the series 1,2,3 converge. We prove that converges as follows. We apply the proposition ( Kolmogorov inequality for series 1 ) to the r.v. for : Since the series 3 converge, the RHS vanishes as : According to the proposition ( Probability based criteria for AS convergence ) this implies that the series converge a.s.

Since the series 2 also converge, we conclude that the series converge a.s.

Note that and the series 1 converge. Hence, the and are equivalent sequences and we already established that the converges a.s. Hence, by the proposition ( Property of equivalent sequences of r.v. ) the converges a.s.

Suppose that the series converge. We prove that 1,2,3 converge as follows. Since converges the cannot be greater then for infinitely many . Hence, By the proposition ( Borel-Cantelli lemma, part 2 ) that the series 1 converge a.s. Hence, the and are equivalent and, by proposition ( Property of equivalent sequences of r.v. ), the series converge. Then the series 2 converge.

To prove that the series 3 converge we apply the proposition ( Kolmogorov inequality for series 2 ) to the r.v. for : If 3 diverges then the RHS above tends to 0 as . Thus, the tail of is not bounded a.s. and such series cannot converge. The contradiction shows that the series 3 must converge.

Proposition

(Equivalence of AS and PR convergence for series) If are independent r.v. then the convergence of a.s. is equivalent to the convergence of in pr.

Proof

Because of the proposition ( AS convergence vs convergence in pr 1 ) it suffices to prove that the convergence in pr implies the convergence a.s.

Hence, we assume that converges in pr: Note that for a hence, We make the LHS sets disjoint if we add the following modification: Therefore, and the is independent from the and . Hence, we continue We conclude, The statement now follows from the proposition ( Probability based criteria for AS convergence ).

 Notation. Index. Contents.