conservation of polymer winding states a combinatoric approach the work in this paper is inspired by a classical problem the statistical physical properties of a closed polymer loop that is wound around a rod historically the preserved topology of this system has been addressed through identification of similarities with magnetic systems we treat the topological invariance in terms of a set of rules that describe all augmentations by additional arcs of some fundamental basic loop of a given winding number these augmentations satisfy the reidemeister move relevant for the polymer with respect to the rod the topologically constrained polymer partition function is now constructed using the combinatorics of allowed arc additions and their appropriate statistical weights we illustrate how for winding number 1 we can formally derive expressions for lower and upper bounds on the partition function using the lower bound approximation we investigate a flexible polymer loop wound between two slits calculating the force on the slit as well as the average numbers of arc types in dependence of slit width and separation results may be extended to higher winding numbers the intuitive nature of this combinatoric scheme allows the development of a variety of approximations and generalizations

marginal densities of the least concave majorant of brownian motion a clean closed form joint density is derived for brownian motion its least concave majorant and its derivative all at the same fixed point some remarkable conditional and marginal distributions follow from this joint density for example it is shown that the height of the least concave majorant of brownian motion at a fixed time point has the same distribution as the distance from the brownian motion path to its least concave majorant at the same fixed time point also it is shown that conditional on the height of the least concave majorant of brownian motion at a fixed time point the left hand slope of the least concave majorant of brownian motion at the same fixed time point is uniformly distributed

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

path integral representation for fractional brownian motion fractional brownian motion fbm is a generalization of the usual brownian motion a path integral representation that has recently been suggested for it is shown to be not for the fbm but for a different generalization of the brownian motion a new path integral representation is given and its measure has fractional derivatives of the path in it the measure shows that the process is gaussian but is in general non markovian even though brownian motion itself is markovian it is shown how the propagator for the motion of free fbm may be evaluated this is somewhat more complex than for the usual path integrals due to the occurrence of fractional derivatives we also find the propagator in the presence of a linear absorption potential and for fbm on a ring

learning a decision boundary from stochastic examples incremental algorithms with and without queries even if it is not possible to reproduce a target input output relation a learning machine should be able to minimize the probability of making errors a practical learning algorithm should also be simple enough to go without memorizing example data if possible incremental algorithms such as error backpropagation satisfy this requirement we propose incremental algorithms that provide fast convergence of the machine parameter to its optimal choice o with respect to the number of examples t we will consider the binary choice model whose target relation has a blurred boundary and the machine whose parameter specifies a decision boundary to make the output prediction the question we wish to address here is how fast can approach o depending upon whether in the learning stage the machine can specify inputs as queries to the target relation or the inputs are drawn from a certain distribution if queries are permitted the machine can achieve the fastest convergence o 2 o t 1 if not o t 1 convergence is generally not attainable for learning without queries we showed in a previous paper that the error minimum algorithm exhibits a slow convergence o 2 o t 2 3 we propose here a practical algorithm that provides a rather fast convergence o t 4 5 it is possible to further accelerate the convergence by using more elaborate algorithms the fastest convergence turned out to be o lnt 2t 1 this scaling is considered optimal among possible algorithms and is not due to the incremental nature of our algorithm

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

new donsker classes several classes of functions are shown to be donsker by an argument based on partitioning the sample space one example is the class of all nondecreasing functions f r r such that 0 f f for a given function f with f 2 dp p

the mean first rotation time of a planar polymer we estimate the mean first time called the mean rotation time mrt for a planar random polymer to wind around a point this polymer is modeled as a collection of n rods each of them being parameterized by a brownian angle we are led to study the sum of i i d imaginary exponentials with one dimensional brownian motions as arguments we find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function finally we obtain an asymptotic formula for the mrt whose leading order term depends on the square root of n and interestingly depends weakly on the mean initial configuration our analytical results are confirmed by brownian simulations our analytical results are confirmed by brownian simulations

mwi representation of the number of curve crossings by a differentiable gaussian process with applications let x x t t 0 be a stationary gaussian process with zero mean continuous spectral distribution and twice differentiable correlation function an explicit representation is given for the number n t of crossings of a c 1 curve by x on the bounded interval 0 t in a multiple wiener ito integral expansion this continues work of the author in which the result was given for 0 the representation is applied to prove new central and noncentral limit theorems for numbers of crossings of constant levels and some consequences for asymptotic variances are given in fixed spectrum settings

the winding angle distribution of an ordinary random walk the winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers the distribution of winding angles is studied here for ordinary or non self avoiding walks new results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution the results of this analysis are compared with simulations of random walks on a lattice the agreement between the two is very good there are however small and as yet unexplained discrepancies

tightness of slip linked polymer chains we study the interplay between entropy and topological constraints for a polymer chain in which sliding rings slip links enforce pair contacts between monomers these slip links divide a closed ring polymer into a number of subloops which can exchange length among each other in the ideal chain limit we find the joint probability density function for the sizes of segments within such a slip linked polymer chain paraknot a particular segment is tight small in size or loose of the order of the overall size of the paraknot depending on both the number of slip links it incorporates and its competition with other segments when self avoiding interactions are included scaling arguments can be used to predict the statistics of segment sizes for certain paraknot configurations

the concave majorant of brownian motion let s t be a version of the slope at time t of the concave majorant of brownian motion on lbrack 0 infty it is shown that the process s 1 s t t 0 is the inverse of a pure jump process with independent nonstationary increments and that brownian motion can be generated by the latter process and brownian excursions between values of the process at successive jump times as an application the limiting distribution of the l 2 norm of the slope of the concave majorant of the empirical process is derived

the behavior of the npmle of a decreasing density near the boundaries of the support we investigate the behavior of the nonparametric maximum likelihood estimator f n for a decreasing density f near the boundaries of the support of f we establish the limiting distribution of f n n where we need to distinguish between different values of 0 a 1 similar results are obtained for the upper endpoint of the support in the case it is finite this yields consistent estimators for the values of f at the boundaries of the support the limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator

semiparametric estimation of the random utility model with rank ordered choice data abstract we propose semiparametric methods for estimating random utility models using rank ordered choice data our primary method is the generalized maximum score gms estimator with partially rank ordered data the gms estimator allows for arbitrary forms of interpersonal heteroskedasticity with fully rank ordered data the gms estimator becomes considerably more flexible allowing for random coefficients and alternative specific heteroskedasticity and correlations the gms estimator has a non standard asymptotic distribution and a convergence rate of n 1 3 we proceed to construct its smoothed version which is asymptotically normal with a faster convergence rate of n d 2 d 1 where d 2 increases in the strength of smoothness assumptions

testing for threshold effects in regression models in this article we develop a general method for testing threshold effects in regression models using sup likelihood ratio lr type statistics although the sup lr type test statistic has been considered in the literature our method for establishing the asymptotic null distribution is new and nonstandard the standard approach in the literature for obtaining the asymptotic null distribution requires that there exist a certain quadratic approximation to the objective function the article provides an alternative novel method that can be used to establish the asymptotic null distribution even when the usual quadratic approximation is intractable we illustrate the usefulness of our approach in the examples of the maximum score estimation maximum likelihood estimation quantile regression and maximum rank correlation estimation we establish consistency and local power properties of the test we provide some simulation results and also an empirical application to tipping in racial segregation this ar

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

optimal cooperation and submodularity for computing potts partition functions with a large number of states the partition function of the q state potts model with random ferromagnetic couplings in the large q limit is generally dominated by the contribution of a single diagram of the high temperature expansion computing this dominant diagram amounts to minimizing a particular submodular function we provide a combinatorial optimization algorithm the optimal cooperation algorithm which works in polynomial time for any lattice the implementation of the method and its running time are also discussed

mean field behavior of the negative weight percolation model on random regular graphs we investigate both analytically and numerically the ensemble of minimum weight loops in the negative weight percolation model on random graphs with fixed connectivity and bimodal weight distribution this allows us to study the mean field behavior of this model the analytical study is based on a conjectured equivalence with the problem of self avoiding walks in a random medium the numerical study is based on a mapping to a standard minimum weight matching problem for which fast algorithms exist both approaches yield results that are in agreement on the location of the phase transition on the value of critical exponents and on the absence of any sizable indications of a glass phase by these results the previously conjectured upper critical dimension of d u 6 is confirmed

maximum likelihood estimation of the survival functions of stochastically ordered random variables abstract many times aspects of populations exist that must satisfy a stochastic ordering requirement nevertheless estimates may not bear out this stochastic ordering because of the inherent variability of the observations this article will consider the problem of finding maximum likelihood estimates of stochastically ordered survival functions for the cases a one survival function being fixed in advance and b estimating two survival functions when the data include censored observations a numerical example is discussed in detail to illustrate the solution to this problem

boundedly pivotal structural change tests in continuous updating gmm with strong weak identification and completely unidentified cases abstract this paper develops structural change tests in the continuous updating gmm framework that are robust to weak identification we propose likelihood ratio like anderson rubin 1949 estimation of the parameters of a single equation in a complete system of stochastic equations annals of mathematical statistics 20 46 63 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university types of tests since the limits of the test statistics are not nuisance parameter free bounds for the limit of the test statistics are derived the bounds are nuisance parameter free and robust to identification problems simulations show that the anderson rubin 1949 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university type of tests have very good small sample properties in the case of weak instruments the sup lm test of andrews 1993a tests for parameter instability and structural change with unknown change point econometrica 61 821 856 rejects the true null of parameter stability more than the nominal level and also has low power in the weak instrument setup

nonparametric tests for constant tail dependence with an application to energy and finance the present paper proposes new tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas to obtain asymptotic properties we derive a new limit result for the sequential empirical tail copula process moreover consistency of both the tests and a change point estimator are proven we analyze the finite sample behavior of the tests by monte carlo simulations finally and crucial from a risk management perspective we apply the new findings to datasets from energy and financial markets

operator formalism for topology conserving crossing dynamics in planar knot diagrams we address here the topological equivalence of knots through the so called reidemeister moves these topology conserving manipulations are recast into dynamical rules on the crossings of knot diagrams this is presented in terms of a simple graphical representation related to the gauss code of knots drawing on techniques for reaction diffusion systems we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded the aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation this necessitates the introduction of composite paulionic operators the formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings subject to topology conserving stochastic dynamics we consider here the simplified situation of two dimensional knot projections however we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes further applicability arises in the context of the simulated annealing of knots the methods presented here depart significantly from the invariant based path integral descriptions often applied in polymer systems and in our view offer a fresh perspective on the conservation of topological states and topological equivalence in knots

algebraic areas enclosed by 2d brownian curves in random media we study closed brownian trajectories when the brownian particle is subjected to a random potential for a poissonian repulsive potential the enclosed algebraic area a is a gaussian variable that scales like t3 4 when t the length of the curve goes to infinity this is intermediate between the situations where the particle is allowed to wander i everywhere on the plane a t and ii only on a bounded domain a t1 2 for the lloyd model we show that the probability distribution p a is the same as in the absence of disorder this surprising result is related to some peculiarities of the cauchy law

maximum score estimation of disequilibrium models and the role of anticipatory price setting abstract in this paper a general model of disequilibrium quantity determination under median regression assumptions is combined with weak assumptions on the price adjustment to yield a consistent maximum score estimator in contrast to sapra 1986 the supply and demand parameters are separately identifiable and the estimator employs all of the observed data the price adjustment assumptions can be motivated by the anticipatory pricing model of green and laffont 1981 in contrast to the green laffont model however our assumptions only require that long run walrasian prices have positive probability as an illustrative application maximum score estimates of a model of the us commercial loan market are computed and compared to maximum likelihood estimates monte carlo results are also reported

two stage plans for estimating the inverse of a monotone function this study investigates two stage plans based on nonparametric procedures for estimating an inverse regression function at a given point specifically isotonic regression is used at stage one to obtain an initial estimate followed by another round of isotonic regression in the vicinity of this estimate at stage two it is shown that such two stage plans accelerate the convergence rate of one stage procedures and are superior to existing two stage procedures that use local parametric approximations at stage two when the available budget is moderate and or the regression function is ill behaved both wald and likelihood ratio type confidence intervals for the threshold value of interest are investigated and the latter are recommended in applications due to their simplicity and robustness the developed plans are illustrated through a comprehensive simulation study and an application to car fuel efficiency data

on the bootstrap of the maximum score estimator this paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator the theory developed also applies to other estimators within a cube root convergence class for some single parameter estimators in this class the results suggest a simple method for inference based upon the bootstrap

persistence of fractional brownian motion with moving boundaries and applications we consider various problems related to the persistence probability of fractional brownian motion fbm which is the probability that the fbm x stays below a certain level until time t recently oshanin et al 2012 arxiv 1209 3313v2 have studied a physical model where persistence properties of fbm are shown to be related to scaling properties of a quantity jn called the steady state current it turns out that for this analysis it is important to determine persistence probabilities of fbm with a moving boundary we show that one can add a boundary of logarithmic order to an fbm without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in oshanin et al 2013 phys rev lett at press arxiv 1209 3313v2 moreover we complement their findings by considering the continuous time version of jt finally we use the results for moving boundaries in order to improve estimates by molchan 1999 commun math phys 205 97 111 concerning the persistence properties of other quantities of interest such as the time when an fbm reaches its maximum on the time interval 0 1 or the last zero in the interval 0 1

polynomial combinatorial optimization methods for analysing the ground states of disordered systems we discuss the application of polynomial combinatorial optimization algorithms to extract the universal zero temperature properties of various disordered systems dijkstras algorithm is used for models of non directed elastic lines on general regular graphs with isotropically correlated random potentials the successive shortest path algorithm for minimum costflow problems is applied for the study of ground state properties and the entanglement of many elastic lines in a disordered environment and the disorder induced loop percolation transition in a vortex glass model the preflow push algorithm for minimum cut maximum flow problems is used for the investigation of a roughening transition occurring in a model for elastic manifolds in a periodic potential in the presence of point disorder

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

windings of random walks l auteur considere une suite de variables aleatoires bornees a valeurs dans r 2 ayant l identite comme matrice de covariance et un vecteur moyen nul il definit une marche aleatoire s a partir des sommes s des 21 premiers termes de la suite et il note n l angle total parcouru par s autour de l origine jusqu a un temps n il demontre sous des conditions de regularite legeres sur la loi du premier element de la suite un resultat sur la convergence de 2 n log n

the distribution of the zeros of random trigonometric polynomials we study the asymptotic distribution of the number z n of zeros of random trigonometric polynomials of degree n as n rightarrow infty it is known that as n grows to infinity the expected number of the zeros is asymptotic to frac 2 sqrt 3 cdot n the asymptotic form of the variance was predicted by bogomolny bohigas and leboeuf to be cn for some c 0 we prove that frac z n bbb e z n sqrt cn converges to the standard gaussian in addition we find that the analogous result is applicable for the number of zeros in short intervals

canonical representatives and equations in hyperbolic groups we use canonical representatives in hyperbolic groups to reduce the theory of equations in torsion free hyperbolic groups to the theory in free groups as a result we get an effective procedure to decide if a system of equations in such groups has a solution for free groups this question was solved by makanin ma and razborov ra the case of quadratic equations in hyperbolic groups has already been solved by lysenok ly our whole construction plays an essential role in the solution of the isomorphism problem for torsion free hyperbolic groups se1 se2

asymptotic theory for nonparametric estimation of survival curves under order restrictions f t g f t all t the nonparametric maximum likelihood estimators in these problems were derived by brunk franck hansen and hogg and dykstra in the present paper we derive their large sample distributions we present two sets of proofs depending on whether or not the data are right censored when centered and scaled by n 1r2 the estimators con verge in distribution to limiting processes related to the concave majorant of brownian motion the limiting distributions are not known in closed form but can be simulated for the purpose of forming asymptotic point wise confidence limits

multiple wiener ito integral expansions for level crossing count functionals this paper applies the stochastic calculus of multiple wiener ito integral expansions to express the number of crossings of the mean level by a stationary discrete or continuous time gaussian process within a fixed time interval 0 t the resulting expansions involve a class of hypergeometric functions for which recursion and differential relations and some asymptotic properties are derived the representation obtained for level crossing counts is applied to prove a central limit theorem of cuzick 1976 for level crossings in continuous time using a general central limit theorem of chambers and slud 1989a for processes expressed via multiple wiener ito integral expansions in terms of a stationary gaussian process analogous results are given also for discrete time processes this approach proves that the limiting variance is strictly positive without additional assumptions needed by cuzick

asymptotic laws for the winding angles of planar brownian motion using constrained path integrals we study the winding angle distribution of a two dimensional brownian motion around a given point by a careful analysis of the spectral properties of some schrodinger like hamiltonians we obtain a generalization of the messulam yor law various limiting cases are considered

numerical results for the ground state interface in a random medium the problem of determining the ground state of a d dimensional interface embedded in a d 1 dimensional random medium is treated numerically using a minimum cut algorithm the exact ground states can be found for a number of problems for which other numerical methods are inexact and slow in particular results are presented for the roughness exponents and ground state energy fluctuations in a random bond ising model it is found that the roughness exponent ensuremath zeta 0 41 ifmmode pm else textpm fi 0 01 0 22 ifmmode pm else textpm fi 0 01 with the related energy exponent being ensuremath theta 0 84 ifmmode pm else textpm fi 0 03 1 45 ifmmode pm else textpm fi 0 04 in d 2 3 respectively these results are compared with previous analytical and numerical estimates

semiparametric identification of binary decision games of incomplete information with correlated private signals abstract this paper studies the identification and estimation of a static binary decision game of incomplete information we make no parametric assumptions on the joint distribution of private signals and allow them to be correlated we show that the parameters of interest can be point identified subject to a scale normalization under mild support requirements for the regressors publicly observed state variables and errors private signals following manski and tamer 2002 we propose a maximum score type estimator for the structural parameters and establish the asymptotic properties of the estimator

subgroup properties of fully residually free groups we prove that fully residually free groups have the howson property that is the intersection of any two finitely generated subgroups in such a group is again finitely generated we also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup

robust inference for threshold regression models abstract this paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump nesting previous works that assume either continuity or discontinuity at the threshold we find that the parameter values under the kink restriction are irregular points of the hessian matrix destroying the asymptotic normality and inducing the cube root convergence rate for the threshold estimate however we are able to obtain the same asymptotic distribution as hansen 2000 for the quasi likelihood ratio statistic for the unknown threshold we propose to construct confidence intervals for the threshold by bootstrap test inversion finite sample performances of the proposed procedures are examined through monte carlo simulations and an economic empirical application is given

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

rates of convergence for estimating regression coefficients in heteroskedastic discrete response models abstract in this paper we consider estimation of discrete response models exhibiting conditional heteroskedasticity of a multiplicative form where the latent error term is assumed to be the product of an unknown scale function and a homoskedastic error term it is first shown that for estimation of the slope coefficients in a binary choice model under this type of restriction the semiparametric information bound is zero even when the homoskedastic error term is parametrically specified hence it is impossible to attain the parametric convergence rate for the parameters of interest however for ordered response models where the response variable can take at least three different values the parameters of interest can be estimated at the parametric rate under the multiplicative heteroskedasticity assumption two estimation procedures are proposed the first estimator based on a parametric restriction on the homoskedastic component of the error term is a two step maximum likelihood estimators where the unknown scale function is estimated nonparametrically in the first stage the second procedure which does not require the parametric restriction estimates the parameters by a kernel weighted least squares procedure under regularity conditions which are standard in the literature both estimators are shown to be n consistent and asymptotically normal

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

optimal third root asymptotic bounds in the statistical estimation of thresholds this paper is concerned with estimating the intersection point of two densities given a sample of both of the densities this problem arises in classification theory the main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size as corollaries we obtain probabilistic bounds for the prediction error in a classification problem the key to the proof is an entropy estimate the lower bounds are based on bounds for general estimators which are applicable in other contexts as well furthermore we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

notes on the two dimensional fractional brownian motion we study the two dimensional fractional brownian motion with hurst parameter h 1 2 in particular we show using stochastic calculus that this process admits a skew product decomposition and deduce from this representation some asymptotic properties of the motion

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

sharp asymptotics for isotonic regression the asymptotic behavior of the isotonic estimator of a monotone regression function that is the least squares estimator under monotonicity restriction is investigated in particular it is proved that the 1 distance between the isotonic estimator and the true function is of magnitude n 1 3 moreover it is proved that a centered version of this 1 distance converges at the n1 2 rate to a gaussian variable with fixed variance

the winding of stationary gaussian processes this paper studies the winding of a continuously differentiable gaussian stationary process f mathbb r rightarrow mathbb c in the interval 0 t we give formulae for the mean and the variance of this random variable the variance is shown to always grow at least linearly with t and conditions for it to be asymptotically linear or quadratic are given moreover we show that if the covariance function together with its second derivative are in l 2 mathbb r then the winding obeys a central limit theorem these results correspond to similar results for zeroes of real valued stationary gaussian functions by malevich cuzick slud and others

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

the asymptotics of s estimators in the linear regression model we consider the consistency and weak convergence of s estimators in the linear regression model sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover for example polynomial trends and i i d carriers a weak convergence theorem for the hampel rousseeuw least median of squares estimator is obtained and it is shown under rather general conditions that the correct norming factor is n 1 3 finally the asymptotic normality of s estimators with a smooth rho function is obtained again under weak conditions on the carriers

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

sle on doubly connected domains and the winding of loop erased random walks two dimensional loop erased random walks lerws are random planar curves whose scaling limit is known to be a schramm loewner evolution sle with parameter 2 in this note some properties of an sle trace on doubly connected domains are studied and a connection to passive scalar diffusion in a burgers flow is emphasised in particular the endpoint probability distribution and winding probabilities for sle2 on a cylinder starting from one boundary component and stopped when hitting the other are found a relation of the result to conditioned one dimensional brownian motion is pointed out moreover this result permits to study the statistics of the winding number for sle2 with fixed endpoints a solution for the endpoint distribution of sle4 on the cylinder is obtained and a relation to reflected brownian motion pointed out

overhangs in interface growth and ground state paths we report on numerical investigations concerning the statistics of the ground state path in a disordered medium for a distribution of energies having a positive support these paths are almost directed even though some overhangs are present computing both the directed minimum energy path and the unconstrained ground state paths allows us to understand whether the two problems belong to the same universality class we present numerical results in dimensions up to d 6 5 1 which suggests that both problems are in the same universality class

critical properties of loop percolation models with optimization constraints we study loop percolation models in two and in three space dimensions in which configurations of occupied bonds are forced to form a closed loop we show that the uncorrelated occupation of elementary plaquettes of the square and the simple cubic lattice by elementary loops leads to a percolation transition that is in the same universality class as the conventional percolation in contrast to this an optimization constraint for the loop configurations which then have to minimize a particular generic energy function leads to a percolation transition that constitutes a universality class for which we report the critical exponents implication for the physics of solid on solid and vortex glass models are discussed

estimation of a panel data sample selection model we consider the problem of estimation in a panel data sample selection model where both the selection and the regression equation of interest contain unobservable individual specific effects we propose a two step estimation procedure which differences out both the sample selection effect and the unobservable individual effect from the equation of interest in the first step the unknown coefficients of the selection equation are consistently estimated the estimates are then used to estimate the regression equation of interest the estimator proposed in this paper is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to n 1 2 depending on the strength of certain smoothness assumptions the finite sample properties of the estimator are investigated in a small monte carlo simulation

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

optimal path in two and three dimensions we apply the dijkstra algorithm to generate optimal paths between two given sites on a lattice representing a disordered energy landscape we study the geometrical and energetic scaling properties of the optimal path where the energies are taken from a uniform distribution our numerical results for both two and three dimensions suggest that the optimal path for random uniformly distributed energies is in the same universality class as the directed polymers we present physical realizations of polymers in a disordered energy landscape for which this result is relevant s1063 651x9808212 9

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

a consistent bootstrap procedure for the maximum score estimator abstract in this paper we propose a new model based smoothed bootstrap procedure for making inference on the maximum score estimator of manski 1975 1985 and prove its consistency we provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem we compare the finite sample performance of different bootstrap procedures through simulation studies the results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes including the m out of n bootstrap additionally we prove a convergence theorem for triangular arrays of random variables arising from binary choice models which may be of independent interest

winding angle distribution of 2d random walks with traps we study analytically the asymptotic behaviour of the average probability for the trajectory of a 2d brownian particle wandering in the presence of randomly distributed traps to wind n times around a given point after a time t it is shown that with where the first exponent represents a well known long time tail of the probability that a particle will not be trapped

winding angle distributions for random walks and flux lines we study analytically and numerically the winding of a flux line around a columnar defect reflecting and absorbing boundary conditions apply to marginal or repulsive defects respectively in both cases the winding angle distribution decays exponentially for large angles with a decay constant depending only on the boundary condition but not on microscopic features nonuniversal distributions are encountered for chiral defects that preferentially twist the flux line in one direction the resulting asymmetric distributions have decay constants that depend on the degree of chirality in particular strong chirality encourages entanglements and leads to broad distributions we also examine the windings of flux lines in the presence of point impurities random bonds our results suggest that pinning to impurities reduces entanglements leading to a narrow gaussian distribution textcopyright 1996 the american physical society

asymptotic normality of statistics based on the convex minorants of empirical distribution functions let f be the uniform empirical distribution function write fn for the least concave majorant of f and let an denote the corresponding density it is shown that n fl fn t 1 2 dt is asymptotically standard normal when centered at log n and normalized by 3 log n 1i2 a similar result is obtained in the 2 sample case in which fn is replaced by the slope of the convex minorant of fm fm hn1

fluctuations of the increment of the argument for the gaussian entire function the gaussian entire function is a random entire function characterised by a certain invariance with respect to isometries of the plane we study the fluctuations of the increment of the argument of the gaussian entire function along planar curves we introduce an inner product on finite formal linear combinations of curves with real coefficients that we call the signed length which describes the limiting covariance of the increment we also establish asymptotic normality of fluctuations

likelihood ratio tests for monotone functions we study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function the likelihood ratio test for this hypothesis is derived in the particular case of interval censoring or current status data and its limiting distribution is obtained the limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving brownian motion t 2 and greatest convex minorants thereof inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function we also study the behavior of the statistic under local and fixed alternatives 1 introduction we shall consider likelihood ratio tests and the corresponding confidence intervals in a class of problems involving nonparametric estimation of a monotone function the problem in each case involves testing the null hypothesis h 0 that the monotone function has a particular value at afixed point t 0 in the domain of the function of course with each testing problem there is a related problem of finding confidence intervals here are some examples of the problems we have in mind

winding number of fractional brownian motion we find the exact winding number distribution of riemann liouville fractional brownian motion for large times in two dimensions using the propagator of a free particle the distribution is similar to the brownian motion case and is of cauchy type in addition we find the winding number distribution of the fractal time process i e the time fractional fokker planck equation in the presence of a finite size winding center

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point