Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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 converges to the random variable in probability (notation in pr.) if

Proposition

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

Proof

Assume the almost sure convergence of to on (see the section ( Operations on sets and logical statements )): We use the set algebra (see formulas ( Intersection property ),( Union property )) to transform the last relationship as follows: According to the above, the . Hence, for any set of the union: for any fixed Note that hence and the claim 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 . We seek indexing such that (see the section ( Operations on sets and logical statements )). Note that The sets are decreasing as increases. Hence, . Therefore, we continue We pick some function that will be determined later and continue Let us now choose so that for some function . We can make such choice because the convergence in probability is given. We obtain Pick and then Hence, for the given choice of independent of for any arbitrarily small . Hence,

Proposition

Almost sure convergence to implies

Proof

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

Proposition

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

Proof

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

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

 Notation. Index. Contents.