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 -additive measure on is called "subprobability measure" (s.p.m.) if .

Definition

(Vague convergence) The sequence of s.p.m. "converges vaguely" to an s.p.m. iff there exists an everywhere dense subset of such that

Notation

We imply if .

Definition

(Continuity point). The number is a "continuity point" of the measure iff , .

Proposition

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

1. , , s.t. , .

2. For every pair of continuity points of

3. vaguely.

Proof

. Let be a pair of continuity points and take any . The sequences and have to converge to by the -additivity of , (see the proposition ( Continuity lemma )). Hence, .

. The set non-continuity numbers of is at most countable because is -additive and . Hence, the set of continuity numbers is dense.

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

Proposition

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

Proof

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

This proves the statement on because, by additivity of , switching from to general alters the difference by no more then .

For we use the condition that are p.m. By (*) and additivity of , Since, and it follows that

Proposition

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

Proof

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

Proposition

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

Proof

A function may be -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 is a sequence of p.m. Then vaguely iff

Definition

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

Proposition

(Precompactness of a tight sequence of p.m.). Let 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 of has a vague limit , where the is s.p.m. To see that is a p.m. observe that tightness implies that for any there are some continuity numbers such that The above is true for any , hence, the is a p.m.

 Notation. Index. Contents.