Fundamental Probability. It details all the essential topics, ranging from standard issues such as order statistics, multivariate normal, and convergence concepts, to more advanced subjects 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.
- 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.
- Devotes a whole chapter to mean–variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions.
- Features a chapter 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.
I Sums of Random Variables.
1 Generating functions.
1.1 The moment generating function.
1.2 Characteristic functions.
1.3 Use of the fast Fourier transform.
1.4 Multivariate case.
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
2.3 Approximating the mean and variance.
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 Xbar and S2 for i.i.d. normal samples.
3.8 Matrix algebra.
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.
4.4 The central limit theorem.
5 Saddlepoint approximations.
5.3 The hypergeometric functions 1F1 and 2F1.
6 Order statistics.
6.1 Distribution theory for i.i.d. samples.
6.2 Further examples.
6.3 Distribution theory for dependent samples.
III More Flexible and Advanced Random Variables.
7 Generalizing and mixing.
7.1 Basic methods of extension.
7.2 Weighted sums of independent random variables.
8 The stable Paretian distribution.
8.1 Symmetric stable.
8.2 Asymmetric stable.
8.5 Generalized central limit theorem.
9 Generalized inverse Gaussian and generalized hyperbolic distributions.
9.2 The modified Bessel function of the third kind.
9.3 Mixtures of normal distributions.
9.4 The generalized inverse Gaussian distribution.
9.5 The generalized hyperbolic distribution.
9.6 Properties of the GHyp distribution family.
10 Noncentral distributions.
10.1 Noncentral chi–square.
10.2 Singly and doubly noncentral F.
10.3 Noncentral beta.
10.4 Singly and doubly noncentral t.
10.5 Saddlepoint uniqueness for the doubly noncentral F.
A Notation and distribution tables.