Enhanced descriptions from Syndetics:

Intermediate Probability is the natural extension of the author's Fundamental Probability. It details several highly important topics, from standard ones such as order statistics, multivariate normal, and convergence concepts, to more advanced ones which are usually not addressed at this mathematical level, or have never previously appeared in textbook form. The author adopts a computational approach throughout, allowing the reader to directly implement the methods, thus greatly enhancing the learning experience and clearly illustrating the applicability, strengths, and weaknesses of the theory.

The book:

Places great emphasis on the numeric computation of convolutions of random variables, via numeric integration, inversion theorems, fast Fourier transforms, saddlepoint approximations, and simulation. Provides introductory material to required mathematical topics such as complex numbers, Laplace and Fourier transforms, matrix algebra, confluent hypergeometric functions, digamma functions, and Bessel functions. Presents full derivation and numerous computational methods of the stable Paretian and the singly and doubly non-central distributions. A whole chapter is dedicated to mean-variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions. A whole chapter is dedicated to nesting, generalizing, and asymmetric extensions of popular distributions, as have become popular in empirical finance and other applications. Provides all essential programming code in Matlab and R.

The user-friendly style of writing and attention to detail means that self-study is easily possible, making the book ideal for senior undergraduate and graduate students of mathematics, statistics, econometrics, finance, insurance, and computer science, as well as researchers and professional statisticians working in these fields.

Table of contents provided by Syndetics

• Preface
• I Sums of Random Variables
• 1 Generating functions
• 1.1 The moment generating function
• 1.1.1 Moments and the m.g.f
• 1.1.2 The cumulant generating function
• 1.1.3 Uniqueness of the m.g.f
• 1.1.4 Vector m.g.f
• 1.2 Characteristic functions
• 1.2.1 Complex numbers
• 1.2.2 Laplace transforms
• 1.2.2.1 Existence of the Laplace transform
• 1.2.2.2 Inverse Laplace transform
• 1.2.3 Basic properties of characteristic functions
• 1.2.4 Relation between the m.g.f. and c.f
• 1.2.4.1 If the m.g.f. exists on a neighbourhood of zero
• 1.2.4.2 The m.g.f. and c.f. for nonnegative X
• 1.2.5 Inversion formulae for mass and density functions
• 1.2.6 Inversion formulae for the c.d.f
• 1.3 Use of the fast Fourier transform
• 1.3.1 Fourier series
• 1.3.2 Discrete and fast Fourier transforms
• 1.3.3 Applying the FFT to c.f. inversion
• 1.4 Multivariate case
• 1.5 Problems
• 2 Sums and other functions of several random variables
• 2.1 Weighted sums of independent random variables
• 2.2 Exact integral expressions for functions of two continuous random variables
• 2.3 Approximating the mean and variance
• 2.4 Problems
• 3 The multivariate normal distribution
• 3.1 Vector expectation and variance
• 3.2 Basic properties of the multivariate normal
• 3.3 Density and moment generating function
• 3.4 Simulation and c.d.f. calculation
• 3.5 Marginal and conditional normal distributions
• 3.6 Partial correlation
• 3.7 Joint distribution of ?X and S2 for i.i.d. normal samples
• 3.8 Matrix algebra
• 3.9 Problems
• II Asymptotics and Other Approximations
• 4 Convergence concepts
• 4.1 Inequalities for random variables
• 4.2 Convergence of sequences of sets
• 4.3 Convergence of sequences of random variables
• 4.3.1 Convergence in probability
• 4.3.2 Almost sure convergence
• 4.3.3 Convergence in r-mean
• 4.3.4 Convergence in distribution
• 4.4 The central limit theorem
• 4.5 Problems
• 5 Saddlepoint approximations
• 5.1 Univariate
• 5.1.1 Density saddlepoint approximation
• 5.1.2 Saddlepoint approximation to the c.d.f
• 5.1.3 Detailed illustration: the normal{{Laplace sum
• 5.2 Multivariate
• 5.2.1 Conditional distributions
• 5.2.2 Bivariate c.d.f. approximation
• 5.2.3 Marginal distributions
• 5.3 The hypergeometric functions 1F1 and 2F1
• 5.4 Problems
• 6 Order statistics
• 6.1 Distribution theory for i.i.d. samples
• 6.1.1 Univariate
• 6.1.2 Multivariate
• 6.1.3 Sample range and midrange
• 6.2 Further examples
• 6.3 Distribution theory for dependent samples
• 6.4 Problems
• III More Flexible and Advanced Random Variables
• 7 Generalizing and mixing
• 7.1 Basic methods of extension
• 7.1.1 Nesting and generalizing constants
• 7.1.3 Extension to the real line
• 7.1.4 Transformations
• 7.1.5 Invention of ?exible forms
• 7.2 Weighted sums of independent random variables
• 7.3 Mixtures
• 7.3.1 Countable mixtures
• 7.3.2 Continuous mixtures
• 7.4 Problems
• 8 The stable Paretian distribution
• Symmetric stable
• 8.2 Asymmetric stable
• 8.3 Moments
• 8.3.1 Mean
• 8.3.2 Fractional absolute moment proof I
• 8.3.3 Fractional absolute moment proof II
• 8.4 Simulation
• 8.5 Generalized central limit theorem
• 9 Generalized inverse Gaussian and generalized hyperbolic distributions
• 9.1 Introduction
• 9.2 The modi?ed Bessel function of the third kind
• 9.3 Mixtures of normal distributions
• 9.3.1 Mixture mechanics
• 9.3.2 Moments and generating functions
• 9.4 The generalized inverse Gaussian distribution
• 9.4.1 De?nition and general formulae
• 9.4.2 The subfamilies of the GIG distribution fam

Author notes provided by Syndetics

Marc S Paolella, Professor of Empirical Finance, Swiss Banking Institute, University of Zurich, Switzerland.