random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

a numerical study of diffusion and conduction in a 2d random medium investigates numerically the diffusion and conduction properties of a two dimensional random lattice hopping model the results are compared with a recent field theoretical renormalisation group approach showing that two is the upper critical dimension for such models the predicted scaling law with a universal logarithmic behaviour of the velocity is clearly observed the finiteness of the diffusion constant are well as its regular dependence versus the strength of randomness are evenly in good agreement with analytical results the author also determines the frequency dependence of the ac conductivity tensor

random walks with feedback on fractal lattices we study numerically a random walk under the competitive processes of a self organized feedback coupling characterized by a strength lambda and an underlying fractal lattice whereas a fractal structure favors a subdiffusive behavior a dynamical feedback leads either to localization in case of an attractive feedback lambda 0 or to superdiffusion for a repulsive memory strength lambda 0 under the influence of both processes the dynamical exponent z is changed for a sierpinski gasket or a sierpinski carpet with repulsive feedback coupling we get 2 z 1 04 or 2 z 1 08 respectively when an attractive feedback is dominant the system offers localization as in the case of a random walk in regular lattices the numerical results are strongly supported by analytical studies based on scaling arguments

growth model for fractal scale free networks generated by a random walk abstract the diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks in the present paper we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale free property if the graph which provides a space where a random walk occurs by itself is formed by the random walk the proposed model is regulated by two parameters p v and p e which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk respectively the power law exponent describing the vertex degree distribution is determined by the ratio p e p v and is related to an internal factor f i via the relation 1 f i 1 where f i is a parameter that describes the local structure generated by the random walk a sufficiently small p v provides the small world property to the model network the small world property is usually considered to be incompatible with the fractal scaling property open m c l c d c where open m c is the average number of vertices which can be reached from a randomly chosen vertex in at most l c steps however we demonstrate that fractality can be reconciled with the small world property by introducing a size dependent fractal cluster dimension d c

crossover in the one dimensional self directed walk the self directed walk is studied in one dimension in this walk with memory the jump probability is given by wn or i 1 exp or g delta n i 1 where delta n is the difference between the number of times the sites in the forward and backward directions have been visited after n steps when g o there is a crossover between a gaussian random walk and an intermediate regime where the radius of gyration grows like n2 followed by a crossover to the asymptotic regime where the walk is directed when g 0 a single crossover is obtained between the gaussian random walk and a saturation regime at large n when the walk is self attracting

self avoiding walks in quenched random environments the self avoiding walk in a quenched random environment is studied using realspace and field theoretic renormalization and flory arguments these methods indicate that the system is described for d de by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path this fixed point is characterized by scaling laws for the size of the walk l n with n the number of steps and the fluctuations in the free energy af l the bound 1 co pore and suggests that this inequality holds for d 2 and 3 although pure cannot be excluded particularly for d 2 for d d c there is a transition between strong and weak disorder phases at which pure the strong disorder fixed point for saws on percolation clusters is discussed the analogy with directed walks is emphasized

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

scaling and crossover in the one dimensional true self avoiding walk the authors apply scaling and crossover arguments to the one dimensional true self avoiding walk which avoids itself with strength g the problem is formulated in terms of a grand canonical ensemble a real space renormalisation group analysis shows that for large repulsion x e g is a relevant variable causing crossover from the self repelling chain src limit with crossover exponent phi 1 a physical interpretation of this result is given in terms of competition between the correlation length and the average distance before the walk turns back on itself the resulting flow suggests the existence of an intermediate attractive fixed point which makes the exponent v different from src and random walk values for all x between 0 and 1 in agreement with recent monte carlo results

rates of convergence of nearest neighbor estimation under arbitrary sampling rates of convergence for nearest neighbor estimation are established in a general framework in terms of metric covering numbers of the underlying space the first result is to find explicit finite sample upper bounds for the classical independent and identically distributed i i d random sampling problem in a separable metric space setting the convergence rate is a function of the covering numbers of the support of the distribution for example for bounded subsets of r sup r the convergence rate is o 1 n sup 2 r the main result is to extend the problem to allow samples drawn from a completely arbitrary random process in a separable metric space and to examine the performance in terms of the individual sample sequences the authors show that for every sequence of samples the asymptotic time average of nearest neighbor risks equals twice the time average of the conditional bayes risks of the sequence finite sample upper bounds under arbitrary sampling are again obtained in terms of the covering numbers of the underlying space in particular for bounded subsets of r sup r the convergence rate of the time averaged risk is o 1 n sup 2 r the authors then establish a consistency result for k sub n nearest neighbor estimation under arbitrary sampling and prove a convergence rate matching established rates for i i d sampling finally they show how their arbitrary sampling results lead to some classical i i d sampling results and in fact extend them to stationary sampling the framework and results are quite general while the proof techniques are surprisingly elementary

markov matrix analysis of random walks in disordered continuous media abstract we study by a markov matrix analysis of the equivalent random walks the dynamic properties of continuous media consisting of both correlated and uncorrelated equal size spheres we employ a blind ant random walk model using the rule that a walker jumps among centers of the directly connected spherical particles on an infinite network the dominant eigenvalues and eigenvectors of the transition probability matrix of the random walks are calculated yielding estimates of the spectral dimension d s and the fractal dimension d w of random walks on the continuous network we find that for the present model the estimates are very close to the corresponding lattice percolation values though only after the finite size effects have been carefully taken into account we also show that the finite size scaling of the largest nontrivial eigenvalues holds for our model with the same exponents as for the lattice percolation

the effect of measurement error summary measurement error causes the distribution generating data to differ from the distribution that is of substantive interest to understand the effect of measurement error on the information produced by statistical procedures it is thus necessary to understand the distortions induced by measurement error this paper sheds light on this by providing small error variance approximations to the distribution of variates subject to contamination by measurement error these provide the basis for the analysis of the properties of estimators and tests and less formal procedures when measurement error is present and the basis for the development of tests to detect measurement error and of models which incorporate its influence

optimal bayes classifiers for functional data and density ratios bayes classifiers for functional data pose a challenge this is because probability density functions do not exist for functional data as a consequence the classical bayes classifier using density quotients needs to be modified we propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified the density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one dimensional density estimates this is an extension to functional data of some of the very earliest nonparametric bayes classifiers that were based on simple density ratios in the one dimensional case by means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect bayes classifiers for functional data can be avoided we demonstrate that in the case of gaussian functional data the proposed functional bayes classifier reduces to a functional version of the classical quadratic discriminant a study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero a phenomenon that has been referred to as perfect classification the proposed classifiers also perform favorably in finite sample applications as we demonstrate in comparisons with other functional classifiers in simulations and various data applications including wine spectral data functional magnetic resonance imaging fmri data for attention deficit hyperactivity disorder adhd patients and yeast gene expression data

statistics of the true self avoiding walk in one dimension numerical results of a monte carlo study for the true self avoiding walk in one dimension are presented for any positive value of the strength parameter 0 g infinity the root mean square displacement rn and the range sn of the walk are characterised by two universal exponents nu 2 3 or 0 003 and s 2 3 or 0 01 respectively for negative g self attracting walk both rn and sn exhibit a saturation effect limn to infinity rn rinfinity g and limn to infinity sn sinfinity g with rinfinity g approximately g 1 and sinfinity g approximately g 1 at g or approximately 0 a simple scaling analysis in n and g is proposed and found to be consistent with the monte carlo results

self avoiding random walk a brownian motion model with local time drift a natural model for a self avoiding brownian motion inr d when specialised and simplified tod 1 becomes the stochastic differential equation x t b t int limits 0 t g x s l s x s ds where l t x t 0 x r is the local time process ofx thoughx is not markovian an analogue of the ray knight theorem holds for l x x r which allows one to prove in many cases of interest that mathop lim limits t to infty x t t exists almost surely and to identify the limit

non universal behaviour of self attracting walks a recently proposed model for self attracting walks is studied using exact enumeration techniques the probability of a step is proportional to exp nu where n 1 for sites already visited by the walker and n 0 for the others with u 0 series for the mean square displacement rn2 approximately n2 nu and the mean number of visited sites sn approximately ns of n step walks are calculated in one to four dimensions in all dimensions anomalous diffusion is observed and exponents nu and s vary continuously with the strength parameter u the results are compared with simulations and with previous results for static and dynamic models of generalized random walks in one dimension the behaviour in two and three dimensions may describe anomalous diffusion in real systems

renormalisation of random walks with memory the renormalisation of field theoretic models of the true self avoiding random walk is analysed for short range interactions the field theory is shown to be renormalisable only in a special case which corresponds to the problem of a random walk in a random environment and non renormalisable otherwise the long range version of the field theory is shown to be renormalisable in a larger region of the parameter space in the case which corresponds to the original long range true self avoiding random walk the renormalisation group analysis is carried out to two loop order anomalous dimensions of various composite operators are calculated and the crossover between scaling regimes described by short range and long range models is discussed

rates of convergence of the functional k nearest neighbor estimate let f be a separable banach space and let x y be a random pair taking values in f r motivated by a broad range of potential applications we investigate rates of convergence of the k nearest neighbor estimate rn x of the regression function r x e y x x based on n independent copies of the pair x y using compact embedding theory we present explicit and general finite sample bounds on the expected squared difference e rn x r x 2 and particularize our results to classical function spaces such as sobolev spaces besov spaces and reproducing kernel hilbert spaces

spiral growing self avoiding walk a new universality class the author introduces the concept of a spiral growing self avoiding walk sgsaw by imposing privman s spiralling constraint ibid vol 16 p l571 1983 on the growing self avoiding walk introduced by majid et al phys rev lett vol 52 p 1257 1984 and lyklema and kremer ibid vol 17 p l691 1984 he studies the critical properties of the sgsaw using monte carlo simulation the exponent nu associated with the average radius of gyration sn is 0 8 on the other hand the critical exponent nu r associated with the end to end distance rn is much less than nu the latter is in sharp contrast to the result nu nu r for the usual saw directed gsaws are shown to behave like directed usual saws

quantile regression with measurement error regression quantiles can be substantially biased when the covariates are measured with error in this paper we propose a new method that produces consistent linear quantile estimation in the presence of covariate measurement error the method corrects the measurement error induced bias by constructing joint estimating equations that simultaneously hold for all the quantile levels an iterative em type estimation algorithm to obtain the solutions to such joint estimation equations is provided the finite sample performance of the proposed method is investigated in a simulation study and compared to the standard regression calibration approach finally we apply our methodology to part of the national collaborative perinatal project growth data a longitudinal study with an unusual measurement error structure

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

quantile regression under misspecification with an application to the u s wage structure quantile regression qr fits a linear model for conditional quantiles just as ordinary least squares ols fits a linear model for conditional means an attractive feature of ols is that it gives the minimum mean squared error linear approximation to the conditional expectation function even when the linear model is misspecified empirical research using quantile regression with discrete covariates suggests that qr may have a similar property but the exact nature of the linear approximation has remained elusive in this paper we show that qr minimizes a weighted mean squared error loss function for specification error the weighting function is an average density of the dependent variable near the true conditional quantile the weighted least squares interpretation of qr is used to derive an omitted variables bias formula and a partial quantile regression concept similar to the relationship between partial regression and ols we also present asymptotic theory for the qr process under misspecification of the conditional quantile function the approximation properties of qr are illustrated using wage data from the u s census these results point to major changes in inequality from 1990 to 2000

understanding the effect of measurement error on quantile regressions the impact of measurement error in explanatory variables on quantile regression functions is investigated using a small variance approximation the approximation shows how the error contaminated and error free quantile regression functions are related a key factor is the distribution of the error free explanatory variable exact calculations probe the accuracy of the approximation the order of the approximation error is unchanged if the density of the error free explanatory variable is replaced by the density of the error contaminated explanatory variable which is easily estimated it is then possible to use the approximation to investigate the sensitivity of estimates to varying amounts of measurement error

instrumental variable treatment of nonclassical measurement error models while the literature on nonclassical measurement error traditionally relies on the availability of an auxiliary data set containing correctly measured observations we establish that the availability of instruments enables the identification of a large class of nonclassical nonlinear errors in variables models with continuously distributed variables our main identifying assumption is that conditional on the value of the true regressors some measure of location of the distribution of the measurement error e g its mean mode or median is equal to zero the proposed approach relies on the eigenvalue eigenfunction decomposition of an integral operator associated with specific joint probability densities the main identifying assumption is used to index the eigenfunctions so that the decomposition is unique we propose a convenient sieve based estimator derive its asymptotic properties and investigate its finite sample behavior through monte carlo simulations

how uniformly a random walker covers a finite lattice we study the distribution of the number of visits a random walker makes at a given site on a finite lattice with n sites during a very long walk which visits each site a large number of times for regular hypercubic lattices in all dimensions we find normal central limit behavior but with anomalously large variance in 2 dimensions in 2 dimensions the ratio of the variance over the average number of visits increases logarithmically with n while it increases n in one dimension we confront this with the case of self repelling also called true self avoiding walks there the variance remains bounded for all times and increases logarithmically with n in 2 dimensions random walks are so ubiquitous in physics and in science in general that one might think that the question how a random walk covers a finite lattice should be well studied it seems that this is not the case in a series of recent papers brummelhuis and hilhorst 1 and others 2 4 have studied a related problem but while they were interested in the distribution of times when the last still unvisited site of the lattice disappears we address in the present paper the behavior at much later times 5 and we ask a different question how large will be the asymptotic fluctuations in the numbers of visits to any given site the only papers where this was studied 6 as far as we are aware of seem obsolete since the authors assumed that the distribution of visits is multifractal it is clear that finally on a uniform lattice with periodic boundary conditions each site will be visited equally often more precisely let us denote by n x the number of times a site x was visited during t steps on a lattice of n sites and let us average over all possible start positions then obviously

nonparametric identification and semiparametric estimation of classical measurement error models without side information virtually all methods aimed at correcting for covariate measurement error in regressions rely on some form of additional information e g validation data known error distributions repeated measurements or instruments in contrast we establish that the fully nonparametric classical errors in variables model is identifiable from data on the regressor and the dependent variable alone unless the model takes a very specific parametric form this parametric family includes but is not limited to the linear specification with normally distributed variables as a well known special case this result relies on standard primitive regularity conditions taking the form of smoothness constraints and nonvanishing characteristic functions assumptions our approach can handle both monotone and nonmonotone specifications provided the latter oscillate a finite number of times given that the very specific unidentified parametric functional form is arguably the exception rather than the rule this identification r

exact series studies of self avoiding walks on two dimensional critical percolation clusters the author presents results of exact series studies of self avoiding walks on percolation clusters on the square lattice performed very close to the percolation threshold and estimates the critical exponents nu and gamma defined by the disorder averages of the mean square radius of gyration or the end to end distance and the number of self avoiding walks respectively the self avoiding walk configurations are enumerated exactly up to 35 steps the result for nu indicates a large increase compared with the self avoiding walks on fully occupied lattice the result for gamma however indicates a behaviour similar to that on a fully occupied lattice the result for nu is completely different from that of lee nakanishi and kim 1989

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

monte carlo renormalisation group for the true self avoiding walk following the general ideas by swendsen 1982 a monte carlo renormalisation group mcrg estimation of the end to end exponent nu for the one dimensional true self avoiding walk tsaw has been carried out because the tsaw is known to reach its asymptotic behaviour only very slowly an approximate expression for the fixed point has been determined by monte carlo techniques this approximate fixed point has been used as a starting point for the mcrg the final result for the exponent nu is 0 665 or 0 004

the generalised true self avoiding walk a model with continuously variable exponent ν the true self avoiding walk is generalised by admitting that the self avoidance parameter depends on the past history of the walk for the generalised model the end to end distance exponent nu is calculated by a method that gives the same result as a flory type theory for the usual true self avoiding walk the exponent nu is predicted to be continuously variable with one of the parameters of the generalised model the resulting expression for the exponent nu is confirmed by monte carlo simulations

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

a unifying model of generalised random walks the authors propose a unifying model for generalised static random walks in which each walk has a weight exp g sigma ni alpha where ni is the occupation number of a site and alpha and g are variable parameters special cases of this general walk include the ordinary brownian random walk the self avoiding walk the lattice domb joyce model 1972 and the interacting random walk model recently introduced by stanley et al 1983 the asymptotic properties of the walk are studied in one dimension by effective medium arguments and exact enumeration methods for repulsive correlations the authors find saw behaviour while for attractive correlations the model is self trapping for 1 alpha or 2 and exhibits anomalous diffusion with continuously varying exponents for 0 or alpha 1 in higher dimensions comments are made about effective medium predictions and their relationship to other generalised random walk models

the nontriviality of the one dimensional problem of a true self avoiding walk it is shown that the root mean square distance r2 12 for the self avoiding walk problem is related to the number of steps n as r2 approximately n2 nu with nu 2 3 rather than the self obvious nu 1 this coincides with the mean field theory result obtained by pietronero 1983 which erroneously was not extended to the case d 1

renormalisation of the true self avoiding walk it is shown that the renormalisation of the true self avoiding walk of amit et al 1982 is its upper critical dimensionality two involves at least two possibly three coupling constants fixed points to first order in epsilon 2 d are identified and the asymptotic behaviour at d 2 is discussed

self attracting walk on lattices we have studied a model of self attracting walk proposed by sapozhnikov using the monte carlo method the mean square displacement and the mean number of visited sites are calculated for one two and three dimensional lattices in one dimension the walk shows diffusive behaviour with however in two and three dimensions we observed a non universal behaviour i e the exponent varies continuously with the strength of the attracting interaction

renormalisation of the long range true self avoiding walk the authors show that the true self avoiding walk with long range interactions leads to a renormalisable field theory they compute the characteristic exponent of the gyration radius up to first order in the difference between the upper critical dimension and space dimensionality and compare it with the estimates based on flory like arguments

robust line estimation with errors in both variables abstract the estimator holding the central place in the theory of the multivariate errors in the variables ev model results from performing orthogonal regression on variables rescaled according to the covariance matrix of the errors our simulations on the univariate model essentially relegate this estimator to use only in large samples on data with no trace of outlier contamination a modification requiring a robust preliminary slope is proposed that essentially sets out the generalization to ev of the robust w estimator in regression it is demonstrated that the modification is robust to outlier contamination even in small samples given a sufficiently good preliminary estimator candidates for a preliminary slope estimator based on the data are discussed and the performance of one under simulation is examined

duration response measurement error the impact of response measurement error in duration data is investigated using small parameter asymptotic approximations and compared with the effect of hazard function heterogeneity the approximations lead to a specification test to detect measurement error which is shown to be related to the class of information matrix tests it is shown that in a commonly used class of models the test statistic is exactly pivotal the second order asymptotic properties of the alternative forms of the test statistic are derived and the quality of the approximations and the performance of the test are investigated via monte carlo experimentation

diffusive limits for true or myopic self avoiding random walks and self repellent brownian polymers in d 3 the problems considered in the present paper have their roots in two different cultures the true or myopic self avoiding walk model tsaw was introduced in the physics literature by amit et al phys rev b 27 1635 1645 1983 this is a nearest neighbor non markovian random walk in mathbb z d which prefers to jump to those neighbors which were less visited in the past the self repelling brownian polymer model srbp initiated in the probabilistic literature by durrett and rogers probab theory relat fields 92 337 349 1992 independently of the physics community is the continuous space time counterpart a diffusion in mathbb r d pushed by the negative gradient of the mollified occupation time measure of the process in both cases similar long memory effects are caused by a path wise self repellency of the trajectories due to a push by the negative gradient of softened local time we investigate the asymptotic behaviour of tsaw and srbp in the non recurrent dimensions first we identify a natural stationary in time and ergodic distribution of the environment the local time profile as seen from the moving particle the main results are diffusive limits in the case of tsaw for a wide class of self interaction functions we establish diffusive lower and upper bounds for the displacement and for a particular more restricted class of interactions we prove full clt for the finite dimensional distributions of the displacement in the case of srbp we prove full clt without restrictions on the interaction functions these results settle part of the conjectures based on non rigorous renormalization group arguments equally valid for the tsaw and srbp cases in amit et al 1983 the proof of the clt follows the non reversible version of kipnis varadhan theory on the way to the proof we slightly weaken the so called graded sector condition

metric properties of fractal lattices the authors study the connectivity dimension d of fractal lattices viewed as networks graphs of sites and constant length bond two examples are investigated in detail the 2d sierpinski gasket and 2d infinite percolation clusters on square and triangular lattices in the first case d is shown to coincide with the fractal dimension whereas d 1 72 or 0 02 appears as a universal exponent for percolation clusters in two dimensions considered as an intrinsic parameter the connectivity dimension d is compared with other intrinsic and extrinsic characteristic parameters of fractal lattices in particular the authors argue that d or d or d holds on fractal lattices in general d fractal dimension d spectral dimension

classical diffusion on eden trees we study an aggregation process which gives rise to compact clusters with no loops a generalization of the node counting theorem applicable to such branched graphs trees is proved this is used to determine the spectral dimension of these clusters on square and cubic embedding lattices by monte carlo simulations the results are explained in terms of the geometric structure of these trees

consistent nonparametric regression for functional data under the stone besicovitch conditions in this paper we address the problem of nonparametric regression estimation in the infinite dimensional setting we start by extending the stone s seminal result to the case of metric spaces when the probability measure of the explanatory variables is tight then under slight variations on the hypotheses we state and prove the theorem for general metric measure spaces from this result we derive the mean square consistency of the k nn and kernel estimators if the regression function is bounded and the besicovitch condition holds we also prove that for the uniform kernel estimate the besicovitch condition is also necessary in order to attain l1 consistency for almost every x

cell renormalisation of growth processes true self avoiding walks and growing animals the authors present a generating function formalism for growth processes in general and introduce a new cell renormalisation scheme as examples they treat the true self avoiding walk and the eden process or growing animals in two and three dimensions using small cells the results in both cases indicate a substantial increase in the fractal dimension d compared with their equilibrium counterparts the case of the eden process may however suggest a peculiar convergence behaviour as the cell size b tends to infinity

approximation of density functions by sequences of exponential families 1 introduction consider the estimation of a probability density func tion p x defined on a bounded interval we approximate the logarithm of the density by a basis function expansion consisting of polynomials splines or trigonometric series the expansion yields a regular exponential family within which we estimate the density by the method of maximum likelihood this method of density estimation arises by application of the principle of maxi mum entropy or minimum relative entropy subject to empirical constraints we show that if the logarithm of the density has r square integrable deriva tives f idr log p12 ox then the sequence of density estimators p converges to p in the sense of relative entropy kullback leibler fp log p pn at rate opr 1 m2r m n as m x c and m2 n 0 in the spline and trigonometric cases and m3 n o 0 in the polynomial case where m is the dimension of the family and n is the sample size boundary conditions are assumed for the density in the trigonometric case this convergence rate specializes to op r n 2r 2r 1 by setting m nl 2r 1 when the log density is known to have degree of smoothness at least r analogous convergence results for the relative entropy are shown to hold in general for any class of log density functions and sequence of finite dimensional linear spaces having l2 and l approximation properties the approximation of log densities using polynomials has previously been considered by neyman 1937 to define alternatives for goodness of fit tests by good 1963 as an application of the method of maximum entropy or minimum relative entropy by crain 1974 1976a b 1977 who demonstrates existence and consistency of the maximum likelihood estimator and by mead and

a short note on the true self avoiding walk the scaling laws for the true self avoiding walk and a fokker planck equation for the end to end probability distribution are combined in order to calculate the scaling functions and the end to end distance in one dimension explicitly

diffusion on a dla cluster in two and three dimensions we present a much improved calculation of the dynamic critical exponents of diffusion dw ds and the combination df dw for the dla diffusion limited aggregate on square and simple cubic lattices using two independent approaches one based on the exact enumeration for the displacement and velocity autocorrelation and the other on the eigenspectrum of random walks on the dla the two methods give consistent results which however clearly rule out the scaling relation ds 2df dw first proposed by alexander and orbach 1982 similarly to the case of the eden tree reported previously

self directed walk a monte carlo study in two dimensions the self directed walk is studied in three dimensions monte carlo simulations on the simple cubic lattice provide an estimate of the radius of gyration exponent nu 0 67 or 0 01 in agreement with the authors recent flory like theory leading to nu 2 d when 2 or d or 4

superdiffusive bounds on self repellent brownian polymers and diffusion in the curl of the gaussian free field in d 2 we consider two models of random diffusion in random environment in two dimensions the first example is the self repelling brownian polymer this describes a diffusion pushed by the negative gradient of its own occupation time measure local time the second example is a diffusion in a fixed random environment given by the curl of massless gaussian free field

on self repelling walks the authors investigate the properties of self repelling walks otherwise known as true self avoiding walks in both one and two dimensions for a range of values of the repulsion parameter g 0 2 or g or 10 0 in one dimension they have obtained 24 terms of the generating function of the mean square end to end distance rn2 while on the two dimensional square lattice they have obtained 12 15 terms in one dimension they find the data to be well fitted by rn2 n43 a b n13 c n o 1 n and in two dimensions by rn2 dn mod lnn mod alpha 1 0 ln mod lnn mod lnn with alpha approximately 0 5 estimates of the amplitudes a and d are also obtained

true self avoiding walks on fractal lattices above the upper marginal dimension we study by monte carlo simulations the critical behaviour and the cross over scaling of the true self avoiding walks tsaw on fractal lattices above the upper marginal dimension we estimate the flory exponent nu which characterizes the rms end to end distances of saw on a percolation cluster at percolation thresholds both in three and four dimensions and on a dla cluster in three dimensions results were in good agreement with the predictions of the known flory formulae we also discuss the fractal to euclidean and the rw to tsaw cross over scaling monte carlo data appear to collapse in the scaling regions for both cases however we found that the scaling function for the latter is different from that on the regular lattices

a comparative study of interacting random walk models the authors analyse four interacting random walk models from a comparative point of view concentrating on one dimensional versions they consider an interacting random walk model recently introduced by stanley et al 1983 the domb joyce model 1972 and the true self avoiding walk of amit et al 1983 in addition they introduce a model based on the weighting of turning points whose properties are related to those of the ising spin chain fixed points and their stability are identified for both attractive and repulsive correlations in all four cases they compare the mechanisms involved in each of these models and comment on the role played in each by its particular form of correlation between steps the different universal behaviours are shown to arise from competition between short range and long range and particularly cumulative memory effects and between local and global normalisation conditions

true self avoiding walks on a percolation cluster the authors investigate by monte carlo simulations the critical behaviour of true self avoiding walks tsaws on a percolation cluster performed very close to the percolation threshold specifically they generate tsaws on a site percolated incipient infinite cluster for various values of the self avoidance parameter g 0 they found that such walks exhibit critical behaviour different from that of ordinary self avoiding walks and also from that of random walks of no constraint the flory exponent obtained was v 0432 or 0 005 for all g 0 which agrees well with the flory type formula suggested by rammal 1984

simulation extrapolation estimation in parametric measurement error models we describe a simulation based method of inference for parametric measurement error models in which the measurement error variance is known or at least well estimated the method entails adding add

series study of the one dimensional true self avoiding walk the true self avoiding walk problem is formulated using a grand canonical approach and exact enumeration methods are used to calculate the average end to end distance for one dimensional true self avoiding walks with up to 21 steps the results are in agreement with a universality picture obtained both from monte carlo simulations and from scaling and crossover arguments the extrapolated value of the end to end distance exponent nu is nu 0 67 or 0 04

monte carlo study of linear diffusion limited aggregation linear diffusion limited aggregation is studied on the square lattice the radius of gyration exponent nu 0 79 or 0 01 is estimated using the monte carlo method and a scaling analysis

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

on the spreading dimension of percolation and directed percolation clusters the authors study the number an of sites that are accessible after n steps at most on clusters at the percolation threshold on a cayley tree an is of order n2 if the origin belongs to a larger cluster whereas its average over all clusters is of order n this suggests that the intrinsic spreading dimension d defined by an approximately nd is equal to two for fractal percolation clusters in space dimensions d or 6 and depends on d for d 6 for directed percolation clusters they argue that d is related to usual critical exponents by d beta gamma nu sub monte carlo data that support this relation are presented in two dimensions analogous results are derived for lattice animals d 2 on the cayley tree and d 1 nu sub for directed animals in any dimensions

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

combinatorial formulae for one dimensional generalised random walks the calculation of ensemble averages in one dimensional walks with weights given by exp g sigma ni alpha where ni is the number of times a site is visited is simplified by reducing the problem to one of calculating binomial factors for partitions and their permutations the combinatorial factors are generalised to the case of cayley trees and all lattices without closed loops in one dimension the combinatorial formulae are shown to reduce the known cases when alpha 0 and alpha 1 the authors also use these formulae to extend the exact enumeration results for the one dimensional generalised random walk from 21 to 29 terms and the series are analysed using pade approximant techniques

random walk statistics on fractal structures we consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds range renewal theory mean first passage time etc asymptotic behaviors are shown to be controlled by the fractal d and spectral d dimensionalities of the considered structure a simple decimation procedure giving the value of d is outlined and illustrated in the case of the sierpinski gaskets recent results for the trapping problem the self avoiding walk and the true self avoiding walk are briefly reviewed new numerical results for diffusion on percolation clusters are also presented

renormalisability and renormalisation of the true self avoiding random walks renormalisation of the recently proposed field theoretic model for the true self avoiding random walk is analysed apart from a special case which corresponds to the problem of a random walk in a random environment the model is shown to be non renormalisable in the case of short range interactions contrary to earlier conjectures the long range version of this field theoretic model is shown to be renormalisable in certain regions of the parameter space and renormalisation group analysis of the model is carried out to two loop order in the region corresponding to the true self avoiding random walk with long range repulsion in a different region of the parameter space the anomalous dimension of the diffusion coefficient is determined exactly in perturbation theory

true self avoiding walk on fractals the authors investigate the true self avoiding walk on fractal spaces the number sn of sites visited during an n step walk and the root mean square displacement rn are calculated via extensive monte carlo simulations on the 2d sierpinski gasket for all positive values of the repulsion parameter g they found the same exponents s 0 815 or 0 001 and nu 0 510 or 0 005 respectively both rn and sn are shown to exhibit a simple scaling behaviour in n and g the authors compare the results with a simple flory like prediction for the exponent nu

from random to self avoiding walks a brief review will be given of the current situation in the theory of self avoiding walks saws the domb joyce model first introduced in 1972 consists of a random walk on a lattice in which eachn step configuration has a weighting factor i 0 n 2 j i 2 n 1 ij herei andj are the lattice sites occupied by the ith and jth points of the walk when 0 the model reduces to a standard random walk and when 1 it is a self avoiding walk the universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function of for largen the implications for the theory of dilute polymer solutions will be indicated

treatment effect estimation with covariate measurement error this paper investigates the effect that covariate measurement error has on a treatment effect analysis built on an unconfoundedness restriction in which there is conditioning on error free covariates the approach uses small parameter asymptotic methods to obtain the approximate effects of measurement error for estimators of average treatment effects the approximations can be estimated using data on observed outcomes the treatment indicator and error contaminated covariates without employing additional information from validation data or instrumental variables the results can be used in a sensitivity analysis to probe the potential effects of measurement error on the evaluation of treatment effects