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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
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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.
A. Operations on sets and logical statements.
B. Fundamental inequalities.
C. Function spaces.
D. Measure theory.
E. Various types of convergence.
a. Uniform convergence and convergence almost surely. Egorov's theorem.
b. Convergence in probability.
c. Infinitely often events. Borel-Cantelli lemma.
d. Integration and convergence.
e. Convergence in Lp.
f. Vague convergence. Convergence in distribution.
F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
G. Lebesgue differentiation theorem.
H. Fubini theorem.
I. Arzela-Ascoli compactness theorem.
J. Partial ordering and maximal principle.
K. Taylor decomposition.
2. Laws 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 in probability.


efinition

The sequence of random variables MATH converges to the random variable $X$ in probability (notation $X_{n}\rightarrow X$ in pr.) if MATH MATH

Proposition

(AS convergence vs convergence in pr 1) Almost sure convergence implies convergence in probability.

Proof

Assume the almost sure convergence of MATH to $X$ on $\Omega$ (see the section ( Operations on sets and logical statements )): MATH We use the set algebra (see formulas ( Intersection property ),( Union property )) to transform the last relationship as follows: MATH According to the above, the MATH . Hence, MATH for any set of the union: for any fixed $m$ MATH Note that MATH hence MATH and the claim MATH follows.

Proposition

(AS convergence vs convergence in pr 2) Convergence in probability implies existence of a subsequence that converges almost surely to the same limit.

Proof

We are given that MATH . We seek indexing MATH such that MATH (see the section ( Operations on sets and logical statements )). Note that MATH The sets MATH are decreasing as $N$ increases. Hence, MATH . Therefore, we continue MATH We pick some function $N\left( m\right) $ that will be determined later and continue MATH Let us now choose MATH so that MATH for some function MATH . We can make such choice because the convergence in probability is given. We obtain MATH Pick MATH and MATH then MATH Hence, MATH for the given choice of MATH independent of $\varepsilon$ for any arbitrarily small $\varepsilon$ . Hence, MATH

Proposition

Almost sure convergence $X_{n}$ to $X$ implies MATH

Proof

We are given MATH Hence, MATH MATH Since the union is a decreasing $N$ -sequence of sets MATH MATH

Proposition

(Probability based criteria for a.s. convergence) Let MATH be any sequence of functions MATH then MATH

Proof

We use the technique of the section ( Operations on sets and logical statements ): MATH Note that MATH We conclude MATH We switched to the sets MATH because these are $N,n,m$ -monotonous. The permits us to use the proposition ( Continuity lemma ).

Indeed, MATH is a decreasing sequence of $M$ and MATH Hence, MATH We repeat this procedure two more times and arrive to the statement of the proposition.





Notation. Index. Contents.


















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