We develop analytic tools for the asymptotics of general trie statistics, which are particularly advantageous for clarifying the asymptotic variance. Many concrete examples are discussed for which new Fourier expansions are given. The tools are also useful for other splitting processes with an underlying binomial distribution. We specially highlight Philippe Flajolet's contribution in the analysis of these random structures.

Dedicated to the 60th birthday of Philippe Flajolet International audience Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n (logn)(2)-variance for certain notions of total path-length is also clarified.

A new approach to Poisson approximation is proposed. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. A selected survey on diverse Poisson approximation results is also given.MSC 2000 Subject Classifications: Primary 62E17; secondary 60C05 60F05.The early history of Poisson approximation. Poisson distribution appeared naturally as the limit of the sum of a large number of independent trials each with very small probability of success. Such a limit form, being the most primitive version of Poisson approximation, dates back to at least de Moivre's work [32] in the early eighteenth century and Poisson's book [61] in the nineteenth century. Haight [38] writes: ". . . although Poisson (or de Moivre) discovered the mathematical expression (1.1-1) [which is e −λ λ k /k!], Bortkiewicz discovered the probability distribution (1.1-1)." And according to Good [37], "perhaps the Poisson distribution should have been named after von Bortkiewicz (1898) because he was the first to write extensively about rare events whereas Poisson added little to what de Moivre had said on the matter and was probably aware of de Moivre's work;" see also Seneta's account in [74] on Abbe's work. In addition to Bortkiewicz's book [17], another important contribution to the early history of Poisson approximation was made by Charlier [21] for his type B expansion, which will play a crucial role in our development of arguments.The next half a century or so after Bortkiewicz and Charlier then witnessed an increase of interests in the properties and applications of the Poisson distribution and Charlier's expansion. In particular, Jordan [47] proved the orthogonality of the Charlier polynomials with respect to the Poisson measure, and considered a formal expansion pair, expressing the Taylor coefficients of a given function in terms of series of Charlier polynomials and vice versa. A sufficient condition justifying the validity of such an expansion pair was later on provided by Uspensky [83]; he also derived very precise estimates for the coefficients in the case of binomial distribution. His complex-analytic approach was later on extended by Shorgin [80] to the more general Poisson-binomial distribution (each trial with a different probability; see next paragraph). Schmidt [73] then gives a sufficient and necessary condition for justifying the Charlier-Jordan expansion; see also Boas [13] and the references therein. Prohorov [65] was the first to study, using elementary arguments, the total variation distance between binomial and Poisson distributions, thus upgrading the classical limit theorem to an approximation theorem.

We consider sequences of random variables whose probability generating functions have only roots on the unit circle, which has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth central and normalized (by the standard deviation) moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance is unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,707–738, 2015

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