Foundations of quantitative finance book IV: distribution functions and expectations

Table of content:
Preface

Introduction

1 Distribution and Density Functions

l.l Summary of Book II Results

l.l.l DistributionFunctionsonJR

l.l.2 Distribution Functions on JRn

l.2 DecompositionofDistributionFunctionsonJR

l.3 DensityFunctionsonJR

l.3.l TheLebesgueApproach

l.3.2 RiemannApproach

l.3.3 Riemann-Stieltjes Framework

l.4 Examples of Distribution Functions on JR

l.4.l DiscreteDistributionFunctions

l.4.2 ContinuousDistributionFunctions

l.4.3 MixedDistributionFunctions

2 Transformed Random Variables-

2.l MonotonicTransformations

2.2 SumsofIndependentRandomVariables

2.2.l DistributionFunctionsofSums

2.2.2 Density Functions of Sums

2.3 Ratios of Random Variables

2.3.l Independent Random Variable

2.3.2 Example without Independence

3 Order Statistics

3.l-M -Samples and Order Statistics

3.2-Distribution Functions for kth Order Statistics

3.3-Density Functions for kth Order Statistics

3.4-Joint Distribution of all Order Statistics

3.5-Density Functions on JRn

3.6-Multivariate Density Functions

-3.6.l Joint Density of all Order Statistics

-3.6.2 Marginal Densities and Distributions

-3.6.3 Conditional Densities and Distributions

3.7-The Renyi Representation Theorem



4 EXpectationsofRandomVariables1

4.l General Definitions

4.l.l Is Expectation Well Defined?

4.l.2 Formal Resolution of Well-Definedness

4.2 Moments of Distributions

4.2.l Common Types of Moments

4.2.2 Moment Generating Function

4.2.3 Moments of Sums - Theory

4.2.4 Moments of Sums - Applications

4.2.5 Properties of Moments

4.2.6 Moment Examples-Discrete Distributions

4.2.7 Moment Examples-Continuous Distributions

4.3 Moment Inequalities

4.3.l Chebyshev's Inequality

4.3.2 Jensen's Inequality

4.3.3 Kolmogorov's Inequality

4.3.4 Cauchy-Schwarz Inequality

4.3.5 Holder and Lyapunov Inequalities

4.4 Uniqueness of Moments

4.4.l Applications of Moment Uniqueness

4.5 Weak Convergence and Moment Limits

5 Simulating Samples of RVs - EXamples

5.l Random Samples

5.l.l Discrete Distributions

5.l.2 Simpler Continuous Distributions

5.l.3 Normal and Lognormal Distributions

5.l.4 Student T Distribution



5.2 Ordered Random Samples

5.2.l Direct Approaches

5.2.2 The Renyi Representation

6 Limit Theorems

6.l Introduction

6.2 Weak Convergence of Distributions

6.2.l Student T ⇒ Normal

6.2.2 Poisson Limit Theorem

6.2.3 "Weak Law of Small Numbers"

6.2.4 De Moivre-Laplace Theorem

6.2.5 The Central Limit Theorem l

6.2.6 Smirnov's Theorem on Uniform Order Statistics

6.2.7 A Limit Theorem on General Quantiles

6.2.8 A Limit Theorem on Exponential Order Statistics

6.3 Laws of Large Numbers

6.3.l Tail Events and Kolmogorov's 0-l Law

6.3.2 Weak Laws of Large Numbers

6.3.3 Strong Laws of Large Numbers

6.3.4 A Limit Theorem in EVT

6.4 Convergence of Empirical Distributions

6.4.l Definition and Basic Properties

6.4.2 The Glivenko-Cantelli Theorem

6.4.3 Distributional Estimates for Dn(s)

7 Estimating Tail Events 2

7.l Large Deviation Theory 2

7.l.l Chernoff Bound

7.l.2 Cramer-Chernoff Theorem

7.2 Extreme Value Theory 2

7.2.l Fisher-Tippett-Gnedenko theorem

7.2.2 The Hill Estimator, 1 > 0

7.2.3 F E D(G,) is Asymptotically Pareto for 1 > 0

7.2.4 F E D(G,), 1 > 0, then 1H � 1

7.2.5 F E D(G,), 1 > 0, then 1H -1 1

7.2.6 Asymptotic Normality of the Hill Estimator
7.2.7 The Pickands-Balkema-de Haan Theorem: 1 > 0

References
[https://www.routledge.com/Foundations-of-Quantitative-Finance-Book-IV-Distribution-Functions-and-Expectations/Reitano/p/book/9781032206523]

Every finance professional wants and needs a competitive edge. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not—and that is the competitive edge these books offer the astute reader.

Published under the collective title of Foundations of Quantitative Finance, this set of ten books develops the advanced topics in mathematics that finance professionals need to advance their careers. These books expand the theory most do not learn in graduate finance programs, or in most financial mathematics undergraduate and graduate courses.

As an investment executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial services industry and two decades in academia where he taught in highly respected graduate programs.

Readers should be quantitatively literate and familiar with the developments in the earlier books in the set. While the set offers a continuous progression through these topics, each title can be studied independently.

(https://www.routledge.com/Foundations-of-Quantitative-Finance-Book-IV-Distribution-Functions-and-Expectations/Reitano/p/book/9781032206523)