We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

In this article, we present a concise review of developments on various continuous multivariate distributions. We first present some basic definitions and notations. Then, we present several important continuous multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
conditional distribution;
truncated distribution;
regression;
bivariate normal;
multivariate normal;
multivariate exponential;
multivariate gamma;
dirichlet;
inverted dirichlet;
liouville;
multivariate logistic;
multivariate pareto;
multivariate extreme value;
multivariate t;
wishart translated systems;
multivariate exponential families

/pdf/continuous-multivariate-distributions-1gnnj8yohf.pdf

Continuous Multivariate Distributions

Here are two books on a topic new to Technometrics: statistical and mathematical demography. The first author of Applied Mathematical Demography wrote the first two editions of this book alone. The second edition was published in 1985. Professor Keyfritz noted in the Preface (p. vii) that at age 90 he had no interest in doing another edition; however, the publisher encouraged him to find a coauthor. The result is an additional focus for the book in the world of biology that makes it much more relevant for the sciences. The book is now part of the publisher’s series on Statistics for Biology and Health. Much of it, of course, focuses on the many aspects of human populations. The new material focuses on mature population models, the particular focus of the new author (see, e.g., Caswell 2000). As one might expect from a book that was originally written in the 1970s, it does not include a lot of information on statistical computing. The new book by Alho and Spencer is focused on putting a better emphasis on statistics in the discipline of demography (Preface, p. vii). It is part of the publisher’s Series in Statistics. The authors are both statisticians, so the focus is on statistics as used for demographic problems. The authors are targeting human applications, so their perspective on science does not extend any further than epidemiology. The book actually strikes a good balance between statistical tools and demographic applications. The authors use the first two chapters to teach statisticians about the concepts of demography. The next four chapters are very similar to the statistics content found in introductory books on survival analysis, such as the recent book by Kleinbaum and Klein (2005), reported by Ziegel (2006). The next three chapters are focused on various aspects of forecasting demographic rates. The book concludes with chapters focusing on three areas of applications: errors in census numbers, financial applications, and small-area estimates.

Univariate Discrete Distributions

Free Random Variables

We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes.

Conditional moments of q-Meixner processes

The product of subsequent partial sums of independent, identically distributed, square integrable, positive random variables is asymptotically lognormal. The result extends in a rather routine way to non-degenerate U-statistics.

https://projecteuclid.org/download/pdf_1/euclid.ecp/1463434772

Asymptotics for Products of Sums and U-statistics

We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a -commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

/pdf/quadratic-harnesses-q-commutations-and-orthogonal-martingale-3rjff19qzy.pdf

Quadratic harnesses, q-commutations, and orthogonal martingale polynomials

Matsumoto and Yor have recently discovered an interesting transformation which preserves a bivariate probability measure which is a product of the generalized inverse Gaussian (GIG) and gamma distributions. This paper is devoted to a detailed study of this phenomenon. Let $X$ and $Y$ be two independent positive random variables. We prove (Theorem 4.1) that $U =(X +Y)^{-1}$ and $V = X^{-1} - (X +Y)^{-1}$ are independent if and only if there exists $p, a, b > 0$ such that $Y$ is gamma distributed with shape parameter $p$ and scale parameter $2 a^-1$, and such that $X$ has a GIG distribution with parameters $-p, a$ and $b$ (the direct part for $a = b$ was obtained in Matsumoto and Yor). The result is partially extended (Theorem 5.1) to the case where $X$ and $Y$ are valued in the cone $V_+$ of symmetric positive definite $(r, r)$ real matrices as follows: under a hypothesis of smoothness of densities, we prove that $U =(X +Y)^-1$ and $V =X^-1 -(X +Y)^ -1$ are independent if and only if there exists $p>(r-1)/2$ and $a$ and $b$ in $V_+$ such that $Y$ is Wishart distributed with shape parameter $p$ and scale parameter $2a^-1$, and such that $X$ has a matrix GIG distribution with parameters $-p, a$ and $b$. The direct result is also extended to singular Wishart distributions (Theorem 3.1).

/pdf/an-independence-property-for-the-product-of-gig-and-gamma-4k7087zooi.pdf

An independence property for the product of GIG and gamma laws

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam-Chihara polynomials and a relation between these polynomials for different values of parameters.

Jacek Wesołowski

Papers

Conditional moments of q-Meixner processes

Asymptotics for Products of Sums and U-statistics

Quadratic harnesses, q-commutations, and orthogonal martingale polynomials

An independence property for the product of GIG and gamma laws

The bi-poisson process: a quadratic harness