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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.

Weak law of large numbers.


(Equivalent sequences of r.v.) The sequences of r.v. MATH are "equivalent" iff MATH


(Property of equivalent sequences of r.v.) If MATH are equivalent then MATH converges a.s. Furthermore if MATH then MATH


It follows from the definition ( Equivalent sequences of r.v. ) and the proposition ( Borel-Cantelli lemma, part 1 ) that MATH We perform the equivalent transformation of the above statement as follows (see the section ( Operations on sets and logical statements )): MATH MATH MATH MATH According to the section ( Operations on sets and logical statements ) we recover the meaning of the above as follows: MATH Such set has the probability 1. Therefore, starting from some $m$ all the terms in the series are zero everywhere except a set of measure 0.


(Law of large number for iid r.v. with finite mean) Let MATH be a family of iid r.v. with a finite mean. Then MATH


Let $F$ be the common distribution function. We have MATH We introduce the variables MATH MATH The MATH are iid. We have MATH According to the proposition ( Estimate of mean by probability series ), MATH and by the proposition ( Property of equivalent sequences of r.v. ) it suffices to show that MATH But such conclusion follows from the proposition ( Simple law of large numbers ) because the MATH have the same finite second moment.


(Law of large numbers for independent r.v.). Let MATH be a sequence of independent r.v. with d.f. MATH . Let MATH is a sequence, MATH , MATH . Suppose

1. MATH as $n\uparrow\infty$

2. MATH as $n\uparrow\infty$ .



The idea of the proof is similar to the previous proposition. Introduce the r.v. MATH Then (1) is equivalent to MATH and (2) is equivalent to MATH Also, MATH

Notation. Index. Contents.

Copyright 2007