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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
A. Zero-or-one laws.
B. Optional random variable. Stopping time.
C. Recurrence of random walk.
D. Fine structure of stopping time.
E. Maximal value of 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.

Zero-or-one laws.


roposition

(Kolmogorov zero-or-one law). For an independent process, each remote event has probability zero or one.

Proof

Let MATH be a remote event and let MATH . We introduce the conditional probability MATH Let MATH . Then by remoteness of MATH and independence of MATH we have MATH Hence, MATH Thus $P_{\Lambda}$ coincides with $P$ on MATH . Consequently, see the proposition ( Random walk space approximation ), $P_{\Lambda}$ coincides with $P$ on MATH . By construction of MATH , MATH We return to the formula MATH with $P_{\Lambda}=P$ : MATH We set $M=\Lambda$ and conclude MATH

Proposition

(Preservation of stationary measure) For a stationary independent process and any MATH we have MATH

Proof

The $\tau^{-1}\Lambda$ is the set $\Lambda$ moved by one position to the right in the coordinate representation. But by stationarity and independence the same measure is assigned to all positions.

Proposition

(Hewitt and Savage zero-or-one law) For a stationary independent process, each permutable set has probability zero or one.

Proof

Let $\Lambda$ be a permutable set. By the proposition ( Random walk space approximation ) there is a sequence MATH such that MATH . For every $k$ there is a permutation $\pi_{k}$ such that MATH . By independence, MATH and by stationarity, MATH By $\sigma$ -additivity of $P$ , (see the proposition ( Continuity lemma )) MATH Hence, we pass the formula $(\ast)$ to the limit and obtain MATH

Proposition

("Infinitely often" zero-or-one law) Let MATH be a sequence of subsets of $\QTR{cal}{R}$ and MATH be a random walk. Then the set MATH is permutable and MATH is equal to zero or one.

Proof

For any $m$ the r.v. $S_{m}$ is invariant with respect to any permutation MATH . Hence, the set MATH is invariant to such permutation $\pi$ . Since the $\Lambda_{m}$ is decreasing as $m\uparrow\infty$ then set MATH is invariant to any permutation MATH for any $n$ . The statement then follows from the proposition ( Hewitt and Savage zero-or-one law ).





Notation. Index. Contents.


















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