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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

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Prediction, learning, and games

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

The set of squares , , is considered and the sum of binary digits is split up into two parts , where collects the first digits and collects the remaining digits. Very precise results on the distribution of and are presented. For example, asymptotic formulae are provided for the numbers and and it is shown that these partial sum-of-digits functions are asymptotically equidistributed in residue classes. These results are prompted by a conjecture by Gelfond saying that the (total) sum-of-digits function is asymptotically equidistributed in residue classes.

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The Sum-of-Digits Function of Squares

We show that the variances of the profile (number of nodes at each level) of random binary search trees undergoes asymptotically four phase transitions and exhibits a bimodal or "two-humped" behavior, in contrast to the unimodality of the expected value of the profiles. Precise asymptotic approximations are derived. The same types of phenomena also hold for the profile of random recursive trees.

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Bimodality and Phase Transitions in the Profile Variance of Random Binary Search Trees

We consider compact group generalizations T(n) of the Thue-Morse sequence and prove that the subsequence T(n2) is uniformly distributed with respect to a measure gv that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group representations.

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Generalized Thue-Morse sequences of squares

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The joint distribution of q-additive functions

Let q be an odd number and Sq,O(n) the difference between the number of k 0 for all n. In this paper it is proved that the same assertion holds if q is divisible by 3 or q = 4N + 1. On the other hand, it is shown that the number of primes q < x with this property is o(x/ log x). Finally, analoga for "higher parities" are provided.

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Michael Drmota

Papers

The Sum-of-Digits Function of Squares

Bimodality and Phase Transitions in the Profile Variance of Random Binary Search Trees

Generalized Thue-Morse sequences of squares

The joint distribution of q-additive functions

Rarified sums of the Thue-Morse sequence