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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.

Vague convergence. Convergence in distribution.


efinition

(Subprobability measure) The $\sigma$ -additive measure $\mu$ on MATH is called "subprobability measure" (s.p.m.) if MATH .

Definition

(Vague convergence) The sequence of s.p.m. MATH "converges vaguely" to an s.p.m. $\mu$ iff there exists an everywhere dense subset $D$ of $\QTR{cal}{R}$ such that MATH

Notation

We imply MATH if $a>b$ .

Definition

(Continuity point). The number $b$ is a "continuity point" of the measure $\mu$ iff $\forall a$ , MATH .

Proposition

(Equivalent definitions of vague convergence). Let MATH be a sequence of s.p.m. and $\mu$ is an s.p.m. The following statements are equivalent.

1. MATH , MATH , MATH s.t. $\forall n>N$ , MATH .

2. For every pair of continuity points $a,b$ of $\mu$ MATH

3. MATH vaguely.

Proof

$1\Rightarrow2$ . Let $a,b$ be a pair of continuity points and take any MATH . The sequences MATH and MATH have to converge to MATH by the $\sigma$ -additivity of $\mu$ , (see the proposition ( Continuity lemma )). Hence, MATH .

$2\Rightarrow3$ . The set non-continuity numbers of $\mu$ is at most countable because $\mu$ is $\sigma$ -additive and MATH . Hence, the set of continuity numbers is dense.

$3\Rightarrow1$ . We assume the contrary, the statement 1 is violated: MATH Then we can construct a subsequence MATH with such property. We have to consider two cases: MATH MATH Fix any dense set $D$ (as a candidate for the definition ( Vague convergence )). In the case (a) there is a pair MATH and MATH . Hence, from the inequality (a) and additivity of $\mu$ and $\mu_{n}$ follows MATH and the statement 3 is violated because the values MATH and MATH are separated by at least $\varepsilon$ .

Proposition

(Uniform property of vague convergence). Let MATH be a sequence of p.m. and $\mu$ is a p.m. If MATH vaguely then MATH

Proof

Fix MATH . There exists a finite set of continuity points MATH such that MATH We apply the proposition ( Equivalent definitions of vague convergence )-2: MATH

This proves the statement on MATH because, by additivity of $\mu,\mu_{n}$ , switching from $a_{k}$ to general $a,b$ alters the difference MATH by no more then MATH .

For MATH we use the condition that MATH are p.m. By (*) and additivity of $\mu ,\mu_{n}$ , MATH Since, MATH and MATH it follows that MATH

Proposition

(Vague precompactness of s.p.m.) For any sequence MATH of s.p.m. there is a subsequence MATH that converges vaguely to a s.p.m.

Proof

Take a dense countable set MATH and let MATH be the counting rule. Denote MATH . The sequence MATH is bounded. Hence, there is a convergent subsequence MATH . The sequence MATH is bounded. Hence, there is a convergent subsequence MATH . We continue so indefinitely and take a diagonal subindexing MATH . The sequence MATH converges at every point MATH . We introduce MATH The $F$ is increasing, right continuous and MATH . Hence, MATH is an s.p.m. and, by construction, MATH

Proposition

(Vague convergence as a weak convergence 1). Let MATH is a sequence of s.p.m. Then MATH vaguely iff MATH

Proof

A function $f\in C_{c}$ may be MATH -approximated by a sequence of piecewise constant function. For the piecewise functions the statement is apparent. This argument works in both directions.

Proposition

(Vague convergence as a weak convergence 2). Let MATH is a sequence of p.m. Then MATH vaguely iff MATH

Proof

See the proofs of the propositions ( Uniform property of vague convergence ) and ( Vague convergence as a weak convergence 1 ).

Definition

(Tight sequence). The sequence of measures MATH is called "tight" if MATH

Proposition

(Precompactness of a tight sequence of p.m.). Let MATH be a tight sequence of p.m. The there is a subsequence that converges vaguely to a p.m..

Proof

According to the proposition ( Vague precompactness of s.p.m. ), some subsequence MATH of MATH has a vague limit $\mu$ , where the $\mu$ is s.p.m. To see that $\mu$ is a p.m. observe that tightness implies that for any $\varepsilon>0$ there are some continuity numbers $a,b$ such that MATH The above is true for any $\varepsilon$ , hence, the $\mu$ is a p.m.





Notation. Index. Contents.


















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