the dual tree of a recursive triangulation of the disk in the recursive lamination of the disk one tries to add chords one after another at random a chord is kept and inserted if it does not intersect any of the previously inserted ones curien and le gall ann probab vol 39 pp 2224 2270 2011 have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process m based on a new approach resembling ideas from the so called contraction method in function spaces we prove that when properly rescaled the planar dual of the discrete lamination converges almost surely in the gromov hausdorff sense to a limit real tree t which is encoded by m this confirms a conjecture of curien and le gall 1 introduction and main results in 18 curien and le gall introduce the model of random recursive triangulations of the disk the construction goes as follows atn 1 two points are sampled independently with uniform distribution on the circle they are connected by a chord a straight line which splits the disk into two fragments later on at each step two independent points are sampled uniformly at random on the circle and are connected by a chord if the latter does not intersect any of the previously inserted chords in other words the two points are connected by a chord if they both fall in the same fragment this gives rise to a sequence of laminations of the disk for us a lamination will be a collection of chords which may only intersect at their end points at timen the lamination ln consists of the union of the chords inserted up to timen as an increasing closed subset of the disk ln converges and it is proved in 18 that

the exponent of discrepancy is at most 1 4778 we study discrepancy with arbitrary weights in the l 2 norm over the d dimensional unit cube the exponent p of discrepancy is defined as the smallest p for which there exists a positive number k such that for all d and all e 1 there exist ke p points with discrepancy at most e it is well known that p 1 2 we improve the upper bound by showing that p 1 4778842 this is done by using relations between discrepancy and integration in the average case setting with the wiener sheet measure our proof is not constructive the known constructive bound on the exponent p is 2 454

the size of random fragmentation trees we consider the random fragmentation process introduced by kolmogorov where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution the dislocation law this is repeated recursively for all pieces more precisely we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold and we study the associated random tree dean and majumdar found a phase transition for this process the number of fragmentations is asymptotically normal for some dislocation laws but not for others depending on the position of roots of a certain characteristic equation this parallels the behavior of discrete analogues with various random trees that have been studied in computer science we give rigorous proofs of this phase transition and add further details the proof uses the contraction method we extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter we believe that this extension has independent interest

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

multilinear pseudodifferential operators beyond calderon zygmund theory abstract we consider two types of multilinear pseudodifferential operators first we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in lebesgue spaces these results generalise earlier work of the present authors concerning linear pseudo pseudodifferential operators secondly we investigate the boundedness of bilinear pseudodifferential operators with symbols in the hormander s m classes these results are new in the case 1 that is outwith the scope of multilinear calderon zygmund theory

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

central limit theorems for the spectra of classes of random fractals we discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal the continuum random tree in the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this though the positivity of the variance arising in the central limit theorem is left open in both cases these fractals can be described through a general crump mode jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known

on a functional contraction method methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space our approach is an extension of the so called contraction method to the space c 0 1 of continuous functions endowed with uniform topology and the space d 0 1 of cadlag functions with the skorokhod topology the contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences it is based on stochastic fixed point equations where probability metrics can be used to obtain contraction properties and allow the application of banach s fixed point theorem we develop the use of the zolotarev metrics on the spaces c 0 1 and d 0 1 in this context applications are given in particular a short proof of donsker s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed

stein s method for diffusion approximations stein s method of obtaining distributional approximations is developed in the context of functional approximation by the wiener process and other gaussian processes an appropriate analogue of the one dimensional stein equation is derived and the necessary properties of its solutions are established the method is applied to the partial sums of stationary sequences and of dissociated arrays to a process version of the wald wolfowitz theorem and to the empirical distribution function

volume growth and heat kernel estimates for the continuum random tree in this article we prove global and local point wise volume and heat kernel bounds for the continuum random tree we demonstrate that there are almost surely logarithmic global fluctuations and log logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r 0 we also show that the on diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t 0 almost surely finally we prove that this quenched almost sure behaviour contrasts with the local annealed averaged over all realisations of the tree volume and heat kernel behaviour which is smooth

weighted norm inequalities for strongly singular convolution operators we derive sharp function estimates for convolution operators whose kernels are more singular than calder6n zygmund kernels this leads to weighted norm inequalities weighted weak 1 1 results are also proved all the results obtained are in the context of ap weights 0 introduction we begin by recalling notations relevant to r 7 given a function f x its fourier transform will be denoted by j 0 ff x e x dx throughout this paper v x and w x will denote nonnegative locally integrable functions we let

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

asymptotic laws for regenerative compositions gamma subordinators and the like for open image in new window a random closed set obtained by exponential transformation of the closed range open image in new window of a subordinator a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of open image in new window we focus on the number of parts k n of the composition when open image in new window is derived from a gamma subordinator we prove logarithmic asymptotics of the moments and central limit theorems for k n and other functionals of the composition such as the number of singletons doubletons etc this study complements our previous work on asymptotics of these functionals when the tail of the levy measure is regularly varying at 0

mellin transforms and asymptotics harmonic sums this survey presents a unified and essentially self contained approach to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics discrete probabilistic models and the average case analysis of algorithms it relies on the mellin transform a close relative of the integral transforms of laplace and fourier the method applies to harmonic sums that are superpositions of rather arbitrary harmonics of a common base function its principle is a precise correspondence between individual terms in the asymptotic expansion of an original function and singularities of the transformed function the main applications are in the area of digital data structures probabilistic algorithms and communication theory

on the rate of convergence in the multivariate clt berry esseen theorems are proved in the multidimensional central limit theorem without using fourier methods an effective and simple estimate of the error in the clt for sums and convex sets using stein s method and induction is derived furthermore the error in the clt for multivariate functions of independent random elements is estimated extending results of van zwet and friedrich to the multivariate case

distributional results for costs of partial match queries in asymmetric k dimensional tries in this paper we study the costs cn of partial match retrievals in k dimensional tries k d tries constructed from n records the probabilistic model that we assume is the asymmetric bernoulli model keys are sequences of independently and identically distributed random variables which assume the values 0 and 1 with probability p neq frac 1 2 and 1 p and are pairwise independent we determine the extremal asymptotic orders that the sequence of expectations mathbb e c n n ge 0 may have for different fixed queries as well as the narrow region that contains mathbb e c n n geq 0 for almost every query furthermore we show that mathbb e c n n geq 0 and rm var c n n ge 0 have the same asymptotics up to a logarithmic factor and employing a central limit theorem for martingale difference arrays we prove asymptotic normality of frac c n mathbb e c n sqrt rm var c n for random queries assumed to be independent of the keys and having their speci

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

on the distribution of betweenness centrality in random trees abstract betweenness centrality is a quantity that is frequently used to measure how central a vertex v is it is defined as the sum over pairs of vertices other than v of the proportions of shortest paths that pass through v in this paper we study the distribution of the betweenness centrality in random trees and related subcritical graph families specifically we prove that the betweenness centrality of the root vertex in a simply generated tree is usually of linear order but has a mean of order n 3 2 we also show that a randomly chosen vertex typically also has linear order betweenness centrality and that the maximum betweenness centrality in a simply generated tree is of order n 2 we obtain limiting distributions for the betweenness centrality of the root vertex and of a randomly chosen vertex as well as for the maximum betweenness centrality and we also show that the centroid has positive probability in the limit to be the vertex of maximum betweenness centrality some similar results also hold for subcritical graph classes which will be briefly discussed finally we study random recursive trees and other families of increasing trees where the situation is quite different here the root betweenness centrality is of quadratic order as is the maximum betweenness centrality the betweenness centrality of a random vertex on the other hand is again of linear order again we also have limiting distributions upon suitable normalisation

weighted sobolev spaces and pseudodifferential operators with smooth symbols absmacr let u i be the fefferman stein sharp function of u and for 1 0 such that the pointwise estimate au x cm u x holds for all x e ri and all schwartz functions u this estimate implies the boundedness of 0 order pseudodifferential operators on weighted lp spaces whenever the weight function belongs to muckenhoupt s class ap having established this we construct weighted sobolev spaces of fractional order in r and on a compact manifold prove a version of sobolev s theorem and exhibit coercive weighted estimates for elliptic pseudodifferential operators

on the exponent of discrepancies we study various discrepancies with arbitrary weights in the l 2 norm over domains whose dimension is proportional to d we are mostly interested in large d the exponent p of discrepancy is defined as the smallest number for which there exists a positive number c such that for all d and e there exist ce p points with discrepancy at most e we prove that for the most standard case of discrepancy anchored at zero the exponent is at most 1 41274 which slightly improves the previously known bound 1 47788 for discrepancy anchored at and for quadrant discrepancy at we prove that the exponent is at most 1 31662 for 1 2 1 2 for unanchored discrepancy we prove that the exponent is at most 1 27113 the previous bound was 1 28898 it is known that for all these discrepancies the exponent is at least 1

the standard additive coalescent regard an element of the set x 1 x 2 x 1 x 2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent of evans and pitman is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j they showed that a version x t t of this process arises as a n weak limit of the process started at time 1 2 log n with n clusters of mass 1 n we show this standard additive coalescent may be constructed from the continuum random tree of aldous by poisson splitting along the skeleton of the tree we describe the distribution of x t on at a fixed time t we show that the size of the cluster containing a given atom as a process in t has a simple representation in terms of the stable subordinator of index 1 2 as t we establish a gaussian limit for centered and normalized cluster sizes and study the size of the largest cluster

squarish k d trees we modify the k d tree on 0 1 d by always cutting the longest edge instead of rotating through the coordinates this modification makes the expected time behavior of lower dimensional partial match queries behave as perfectly balanced complete k d trees on n nodes this is in contrast to a result of flajolet and puech j assoc comput mach 33 1986 pp 371 407 who proved that for standard random k d trees with cuts that rotate among the coordinate axes the expected time behavior is much worse than for balanced complete k d trees we also provide results for range searching and nearest neighbor search for our trees

fast implementations of nearest neighbor classifiers abstract standard implementations of non parametric classifiers have large computational requirements parzen classifiers use the distances of an unknown vector to all n prototype samples and consequently exhibit o n behavior in both memory and time we describe four techniques for expediting the nearest neighbor methods replacing the linear search with a new kd tree method exhibiting approximately o n 1 2 behavior employing an l instead of l 2 distance metric using variance ordered features and rejecting prototypes by evaluating distances in low dimensionality subspaces we demonstrate that variance ordered features yield significant efficiency gains over the same features linearly transformed to have uniform variance we give results for a large ocr problem but note that the techniques expedite recognition for arbitrary applications three of four techniques preserve recognition accuracy

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

partial match queries in random k d trees we solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random k d trees our approach is new and of some generality in particular it is applicable to many problems involving differential equations or difference equations with polynomial coefficients

integration and approximation of multivariate functions average case complexity with isotropic wiener measure we study the average case complexity of multivariate integration and l 2 function approximation for the class f c 0 1 d of continuous functions of d variables the class f is endowed with the isotropic wiener measure brownian motion in levy s sense furthermore for both problems only function values are used as data

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

an analysis of random d dimensional quad trees it is shown that the depth of the last node inserted in a random quad tree constructed from independent uniform 0 1 d random vectors is in probability asymptotic to 2 d log n where log denotes the natural logarithm in addition for d 2 exact values are obtained for all the moments of the depth of the last node

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

statistical tests for large tree structured data we develop a general statistical framework for the analysis and inference of large tree structured data with a focus on developing asymptotic goodness of fit tests we first propose a consistent statistical model for binary trees from which we develop a class of invariant tests using the model for binary trees we then construct tests for general trees by using the distributional properties of the continuum random tree which arises as the invariant limit for a broad class of models for tree structured data based on conditioned galton watson processes the test statistics for the goodness of fit tests are simple to compute and are asymptotically distributed as 2 and f random variables we illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer we use a novel approach with tree based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

on the contraction method with degenerate limit equation a class of random recursive sequences yn with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form x stackrel mathcal l x for nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the unique solution of the limit equation in this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation in particular we establish some new tools and a general convergence scheme which transfers information on mean and variance into a central limit law with normal limit we also obtain a convergence rate result for the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables

partial match retrieval of multidimensional data a precise analysis of partial match retrieval of multidimensional data is presented the structures considered here are multidimensional search trees k d trees and digital tries k d tries as well as structures designed for efficient retrieval of information stored on external devices the methods used include a detailed study of a differential system around a regular singular point in conjunction with suitable contour integration techniques for the analysis of k d trees and properties of the mellin integral transform for k d tries and extendible cell algorithms

stability of the bakry emery theorem on rn abstract we establish quantitative stability estimates for the bakry emery bound on logarithmic sobolev and poincare constants of uniformly log concave measures more specifically we show that if a 1 uniformly log concave measure has almost the same logarithmic sobolev or poincare constant as the standard gaussian measure then it almost splits off a gaussian factor our results are dimension free leading to dimension free stability estimates for gaussian concentration of lipschitz functions the proofs are based on stein s method optimal transport and an approximate integration by parts identity relating measures and near extremals in the associated functional inequality

mellin transforms and asymptotics finite differences and rice s integrals high order differences of simple number sequences may be analysed asymptotically by means of integral representations residue calculus and contour integration this technique akin to mellin transform asymptotics is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries digital search trees quadtrees and distributed leader election

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

weighted norm inequalities for pseudo pseudodifferential operators defined by amplitudes abstract we prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables the result is sharp even for smooth amplitudes nevertheless in the case when the amplitude contains the oscillatory factor e i 1 the result can be substantially improved we extend the l p boundedness of pseudo pseudodifferential operators to certain weights end point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation