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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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 MATH , MATH be sequences of real numbers, MATH . If the series MATH converge then MATH , MATH .

Proof

We set MATH Observe that MATH MATH We make a change $i=j-1$ in the second term: MATH Since $a_{j}-a_{j+1}\leq0$ we can use the mean value theorem MATH where the MATH is a "middle" value with the property MATH Note that MATH Hence, MATH

Proposition

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

1. MATH ,

2. MATH , $\forall x$ ,

3. MATH is increasing,

4. MATH is decreasing.

Let MATH be a sequence of independent r.v. with MATH for every $n$ . Let MATH be a sequence of real numbers, $a_{n}>0$ , MATH . If $\ $ MATH then MATH and MATH

Proof

We denote $F_{n}$ the d.f. of the r.v. $X_{n}$ .

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

a. MATH ,

b. MATH ,

c. MATH

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

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

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

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

Proposition

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

Proof

We define MATH We have MATH According to the proposition ( Estimate of mean by probability series ) the last series converge. Hence, $X_{n}$ and $Y_{n}$ 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 MATH . To accomplish it we apply the proposition ( Strong law of large numbers for mean zero ) to the sequence MATH and MATH . We estimate MATH Here the $F$ is the d.f. of $X_{1}$ . MATH The term MATH may be estimated using MATH . The result is MATH for some constant $j$ . In addition, MATH on the set MATH . MATH





Notation. Index. Contents.


















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