rigidity and tolerance in point processes gaussian zeros and ginibre eigenvalues let be a translation invariant point process on the complex plane c and let d c be a bounded open set we ask the following what does the point configuration out obtained by taking the points of outside d tell us about the point configuration in of inside d we show that for the ginibre ensemble out determines the number of points in in for the translation invariant zero process of a planar gaussian analytic function we show that out determines the number as well as the center of mass of the points in in further in both models we prove that the outside says nothing more about the inside in the sense that the conditional distribution of the inside points given the outside is mutually absolutely continuous with respect to the lebesgue measure on its supporting submanifold

invariant palm and related disintegrations via skew factorization let g be a measurable group with haar measure acting properly on a space s and measurably on a space t then any finite jointly invariant measure m on s t admits a disintegration nu otimes mu into an invariant measure on s and an invariant kernel from s to t here we construct and by a general skew factorization which extends an approach by rother and zahle for homogeneous spaces s over g this leads to easy extensions of some classical propositions for invariant disintegration previously known in the homogeneous case the results are applied to the palm measures of jointly stationary pairs where is a random measure on s and is a random element in t

skorokhod embeddings for two sided markov chains let x n n in mathbb z be a two sided recurrent markov chain with fixed initial state x 0 and let nu be a probability measure on its state space we give a necessary and sufficient criterion for the existence of a non randomized time t such that x t n n in mathbb z has the law of the same markov chain with initial distribution nu in the case when our criterion is satisfied we give an explicit solution which is also a stopping time and study its moment properties we show that this solution minimizes the expectation of psi t in the class of all non negative solutions simultaneously for all non negative concave functions psi

transforming random elements and shifting random fields consider a locally compact second countable topological transformation group acting measurably on an arbitrary space we show that the distributions of two random elements x and x in this space agree on invariant sets if and only if there is a random transformation r such that rx has the same distribution as x applying this to random fields in d dimensions under site shifts we show further that these equivalent claims are also equivalent to site average total variation convergence this convergence result extends to amenable groups

extra heads and invariant allocations let n be an ergodic simple point process on e d and let n be its palm version thorisson ann probab 24 1996 2057 2064 proved that there exists a shift coupling of and n that is one can select a random point y of n such that translating n by y yields a configuration whose law is that of we construct shift couplings in which y and are functions of and prove that there is no shift coupling in which n is a function of the key ingredient is a deterministic translation invariant rule to allocate sets of equal volume forming a partition of r d to the points of n the construction is based on the gale shapley stable marriage algorithm amer math monthly 69 1962 9 15 next let be an ergodic random element of 0 1 zd and let be r conditioned on r 0 1 a shift coupling x of r and is called an extra head scheme we show that there exists an extra head scheme which is a function of r if and only if the marginal e r 0 is the reciprocal of an integer when the law of is product measure and d 3 we prove that there exists an extra head scheme x satisfying e exp c x d this answers a question of holroyd and liggett ann probab 29 2001 1405 1425

the total coordinate ring of a smooth projective surface in 5 cox introduced the homogeneous coordinate ring of a toric variety which is a polynomial ring that allows to show that such a variety behaves like a projective space in many ways an analogous to that ring can also be defined for a smooth projective variety x over an algebraically closed field k such that linear and numerical equivalence coincide for divisors on x condition which is assumed for all varieties considered in this paper indeed let us fix li i 1 a z basis of pic x and set n n1 n2 nr z and dn ri 1 nili by regarding the vector spaces h 0 x ox dn f k divx f dn 0 0 as k subvector spaces of the function field k of x the total coordinate ring of x or the cox ring of x is defined as the graded k subalgebra of k

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

gravitational allocation to poisson points for d 3 we construct a non randomized fair and translationequivariant allocation of lebesgue measure to the points of a standard poisson point process in r d defined by allocating to each of the poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the poisson process we prove that this allocation rule is economical in the sense that the allocation diameter defined as the diameter x of the basin of attraction containing the origin is a random variable with a rapidly decaying tail specifically we have the tail bound p x r c exp h cr logr d i

invariant transports of stationary random measures and mass stationarity we introduce and discuss balancing invariant transports of stationary random measures on a polish abelian group the first main result is an associated fundamental invariance property of palm measures derived from a generalization of neveu s exchange formula the second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transports we then introduce in a non stationary setting the concept of mass stationarity with respect to a random measure formalizing the intuitive idea that the origin is a typical location in the mass the third main result of the paper is that a measure is a palm measure if and only if it is mass stationary

poisson splitting by factors given a homogeneous poisson process on d with intensity we prove that it is possible to partition the points into two sets as a deterministic function of the process and in an isometry equivariant way so that each set of points forms a homogeneous poisson process with any given pair of intensities summing to in particular this answers a question of ball electron commun probab 10 2005 60 69 who proved that in d 1 the poisson points may be similarly partitioned via a translation equivariant function so that one set forms a poisson process of lower intensity and asked whether the same is possible for all d we do not know whether it is possible similarly to add points again chosen as a deterministic function of a poisson process to obtain a poisson process of higher intensity but we prove that this is not possible under an additional finitariness condition

unbiased shifts of brownian motion let b b t t r be a two sided standard brownian motion an unbiased shift of b is a random time t which is a measurable function of b such that b t t b t t r is a brownian motion independent of b t we characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of b for any probability distribution on r we construct a stopping time t 0 with the above properties such that b t has distribution we also study moment and minimality properties of unbiased shifts a crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on r another new result is an analogue for diffuse random measures on r of the cycle stationarity characterisation of palm versions of stationary simple point processes

the cone conjecture for calabi yau pairs in dimension 2 a central idea of minimal model theory as formulated by mori is to study algebraic varieties using convex geometry the cone of curves of a projective variety is defined as the convex cone spanned by the numerical equivalence classes of algebraic curves the dual cone is the cone of nef line bundles for fano varieties varieties with ample anticanonical bundle these cones are rational polyhedral by the cone theorem 21 theorem 3 7 for more general varieties these cones are not well understood they can have infinitely many isolated extremal rays or they can be round both phenomena occur among calabi yau varieties such as k3 surfaces 22 which can be considered the next simplest varieties after fano varieties the morrison kawamata cone conjecture would give a clear picture of the nef cone for calabi yau varieties 29 30 19 the conjecture says that the action of the automorphism group of the variety on the nef cone has a rational polyhedral fundamental domain the conjecture includes an analogous statement about the movable cone see section 2 for details thus for calabi yau varieties the failure of the nef cone to be rational polyhedral is always explained by an infinite discrete group of automorphisms of the variety it is not clear where these automorphisms should come from nonetheless the conjecture has been proved for calabi yau surfaces by sterk looijenga and namikawa 41 33 19 the heart of the proof being the torelli theorem for k3 surfaces of piatetski shapiro and shafarevich 3 theorem 11 1 kawamata proved the cone conjecture for all 3 dimensional calabi yau fiber spaces over a positive dimensional base 19 the conjecture is wide open for calabiyau 3 folds despite significant results by oguiso and peternell 38 szendroi 43 uehara 46 and wilson 48 the conjecture was generalized from calabi yau varieties to klt calabi yau pairs x in 45 here is a divisor on x and calabi yau means that kx rather than kx is numerically trivial in this paper we prove the cone conjecture for all klt calabi yau pairs of dimension 2 theorem 4 1 using the geometry of groups acting on convex cones and reduction to the case of k3 surfaces this is enough to show that the conjecture is reasonable in the greater generality of pairs more concretely the theorem gives control over the nef cone and the automorphism group for a large class of rational surfaces including the fano surfaces as well as many others in particular we get a good description of when the cox ring or total coordinate ring is finitely generated in this class of surfaces corollary 5 1 thanks to caucher birkar igor dolgachev brian harbourne artie prendergastsmith chenyang xu and the referees for their comments

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression abstract broad classes of diagnostics for serial correlation and or dynamic conditional heteroskedasticity of regression disturbances are considered the classes include statistics with good power against strongly dependent alternatives along with the usual ones that test against weak dependence and many others limiting null distributions are obtained under mild conditions on the dependence structure of the alternative against which the test is derived on moments of the disturbances and on the regressors the various test statistics have a similar overall structure and while tests against strongly dependent alternatives entail more computation than ones against weakly dependent alternatives the difference can be slight if the fast fourier transform is used

a stable marriage of poisson and lebesgue let e be a discrete set in r d call the elements of e centers the well known voronoi tessellation partitions r d into polyhedral regions of varying sizes by allocating each site of r d to the closest center here we study fair allocations of r d to e in which the regions allocated to different centers have equal volumes we prove that if e is obtained from a translation invariant point process then there is a unique fair allocation which is stable in the sense of the galeshapley marriage problem i e sites and centers both prefer to be allocated as close as possible and an allocation is said to be unstable if some site and center both prefer each other over their current allocations we show that the region allocated to each center is a union of finitely many bounded connected sets however in the case of a poisson process an infinite volume of sites are allocated to centers further away than we prove power law lower bounds on the allocation distance of a typical site it is an open problem to prove any upper bound in d 1

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

minkowski decomposition of okounkov bodies on surfaces abstract we prove that the okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo effective cone is rational polyhedral decomposes as the minkowski sum of finitely many simplices and line segments arising as okounkov bodies of nef divisors

gravitational allocation on the sphere given a collection l of n points on a sphere s n 2 of surface area n a fair allocation is a partition of the sphere into n parts each of area 1 and each is associated with a distinct point of l we show that if the n points are chosen uniformly at random and if the partition is defined by a certain gravitational potential then the expected distance between a point on the sphere and the associated point of l is o log n we use our result to define a matching between two collections of n independent and uniform points on the sphere and prove that the expected distance between a pair of matched points is o log n which is optimal by a result of ajtai komlos and tusnady

detecting changes in the mean of functional observations principal component analysis has become a fundamental tool of functional data analysis it represents tne functional data as x i t t 1 i l v l t wnere is the common mean v l are the eigenfunctions of the covariance operator and the i l are the scores inferential procedures assume that the mean function t is the same for all values of i if in fact the observations do not come from one population but rather their mean changes at some point s the results of principal component analysis are confounded by the change s it is therefore important to develop a methodology to test the assumption of a common functional mean we develop such a test using quantities which can be readily computed in the r package fda the null distribution of the test statistic is asymptotically pivotal with a well known asymptotic distribution the asymptotic test has excellent finite sample performance its application is illustrated on temperature data from england

on the cone of curves of an abelian variety l etx be a smooth projective variety over the complex numbers and let n1 x b e the real vector space of 1 cycles on x modulo numerical equivalence as usual denote by ne x t he cone of curves on x i e the convex cone in n1 x generated by the effective 1 cycles one knows by the cone theorem 4 that the closed cone of curves ne x is rational polyhedral whenever c1 x is ample for varieties x such that c1 x is not ample however it is in general difficult to determine the structure of ne x the purpose of this paper is to study the cone of curves of abelian varieties specifically the abelian varieties x are determined such that the closed cone ne x i s rational polyhedral the result can also be formulated in terms of the nef cone of x or in terms of the semi group of effective classes in the neron severi group of x

counting zariski chambers on del pezzo surfaces abstract zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series in the present paper we present an algorithm that allows to effectively determine zariski chambers when the negative curves on the surface are known we show how the algorithm can be used to compute the number of chambers on del pezzo surfaces

optimal transport from lebesgue to poisson this paper is devoted to the study of couplings of the lebesgue measure and the poisson point process we prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite moreover we give precise conditions for the latter which demonstrate a sharp threshold at d 2 the cost will be defined in terms of an arbitrary increasing function of the distance the coupling will be realized by means of a transport map allocation map which assigns to each poisson point a set cell of lebesgue measure 1 in the case of quadratic costs all these cells will be convex polytopes

weakly dependent functional data functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals for example days the functions thus obtained form a functional time series and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data for scalar and vector valued stochastic processes a large number of dependence notions have been proposed mostly involving mixing type distances between algebras in time series analysis measures of dependence based on moments have proven most useful autocovariances and cumulants we introduce a moment based notion of dependence for functional time series which involves m dependence we show that it is applicable to linear as well as nonlinear functional time series then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data we study the estimation of the functional principal components the long run covariance matrix change point detection and the functional linear model we explain when temporal dependence affects the results obtained for i i d functional observations and when these results are robust to weak dependence

on discriminating between long range dependence and changes in mean we develop a testing procedure for distinguishing between a long range dependent time series and a weakly dependent time series with change points in the mean in the simplest case under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point and under the alternative it is long range dependent we compute the cusum statistic t n which allows us to construct an estimator k of a change point we then compute the statistic t n 1 based on the observations up to time k and the statistic t n 2 2 based on the observations after time k the statistic m n max t n 1 t n 2 converges to a well known distribution under the null but diverges to infinity if the observations exhibit long range dependence the theory is illustrated by examples and an application to the returns of the dow jones index

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

how to find an extra head optimal random shifts of bernoulli and poisson random fields we consider the following problem given an i i d family of bernoulli random variables indexed by z d find a random occupied site x z d such that relative to x the other random variables are still i i d bernoulli results of thorisson imply that such an x exists for all d liggett proved that for d 1 there exists an x with tails p x t of order ct 1 2 but none with finite 1 2th moment we prove that for general d there exists a solution with tails of order ct d 2 while for d 2 there is none with finite first moment we also prove analogous results for a continuum version of the same problem finally we prove a result which strongly suggests that the tail behavior mentioned above is the best possible for all d