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.

## Strong law of large numbers.

roposition

(Kronecker summation lemma) Let , be sequences of real numbers, . If the series converge then , .

Proof

We set Observe that We make a change in the second term: Since we can use the mean value theorem where the is a "middle" value with the property Note that Hence,

Proposition

(Strong law of large numbers for mean zero). Assume that the function satisfies the following criteria

1. ,

2. , ,

3. is increasing,

4. is decreasing.

Let be a sequence of independent r.v. with for every . Let be a sequence of real numbers, , . If then and

Proof

We denote the d.f. of the r.v. .

We aim to apply the proposition ( Kolmogorov three series theorem ) to the series . Hence, we introduce the r.v. and proceed to verify that the series

a. ,

b. ,

c.

are all convergent. The last conclusion then follows from the proposition ( Kronecker summation lemma ).

(a). According to the condition 3 the function is increasing, hence . We continue

(b). Note that . Hence, . According to the condition 3 the function is increasing and according to 1 . Hence . We continue

(c). According to the condition 4 the function is decreasing. Hence . We continue

Proposition

(Strong law of large numbers for iid r.v.) Let be a sequence of iid r.v. Then

Proof

We define We have According to the proposition ( Estimate of mean by probability series ) the last series converge. Hence, and are equivalent sequences and, according to the proposition ( Property of equivalent sequences of r.v. ), it suffices to prove the first part of the proposition for . To accomplish it we apply the proposition ( Strong law of large numbers for mean zero ) to the sequence and . We estimate Here the is the d.f. of . The term may be estimated using . The result is for some constant . In addition, on the set .

 Notation. Index. Contents.