aging of asymmetric dynamics on the random energy model we show aging of glauber type dynamics on the random energy model in the sense that we obtain the annealed scaling limits of the clock process and of the age process the latter encodes the gibbs weight of the configuration occupied by the dynamics both limits are expressed in terms of stable subordinators

the universal relation between scaling exponents in first passage percolation it has been conjectured in numerous physics papers that in ordinary rst passage percolation on integer lattices the uctuation exponent and the wandering exponent are related through the universal relation 2 1 irrespective of the dimension this is sometimes called the kpz relation between the two exponents this article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense

homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

local tail bounds for functions of independent random variables it is shown that functions defined on 0 1 r 1 n satisfying certain conditions of bounded differences that guarantee sub gaussian tail behavior also satisfy a much stronger local sub gaussian property for self bounding and configuration functions we derive analogous locally subexponential behavior the key tool is talagrand s ann probab 22 1994 1576 1587 variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on 0 1 r 1 n for r 2

eden growth on multifractal lattices a modified eden model has been investigated in which the growth probabilities are determined by a fractal measure on the underlying lattice both the spatial distribution and probability distribution of surface sites on the growing clusters have been investigated for the case where the multifractal substrate is constructed using a 2 2 multiplicative generator with the probabilities p1 1 p2 r p3 r3 and p4 r3 the spatial distribution of surface sites and the sites comprising the inner and outer hulls have a fractal geometry which can be described by dimensionalities which depend on r for the total surface this dimensionality converges to a value of about 1 76 or 0 01 as r to 0 for the inner and outer hulls the fractal dimensionality approaches a value of about 1 48 or 0 02

a law of large numbers for nearest neighbour statistics in practical data analysis methods based on proximity near neighbour relationships between sample points are important because these relations can be computed in time n log n as the number of points n associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample if the sample points are independent and identically distributed the associated random variables will also be identically distributed but not independent despite this we show that random variables of this type satisfy a strong law of large numbers in the sense that their sample means converge to their expected values almost surely as the number of sample points n

large deviations for increasing sequences on the plane we prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among poisson points on the plane the rate function for lower tail deviations is derived from a 1977 result of logan and shepp about young diagrams of random permutations for the upper tail we use a coupling with hammersley s particle process and convex analytic techniques along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric hammersley s process

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

classification of homogeneous data with large alphabets given training sequences generated by two distinct but unknown distributions on a common alphabet we study the problem of determining whether a third sequence was generated according to the first or second distribution to model sources such as natural language for which the underlying distributions are difficult to learn from realistic amounts of data we allow the alphabet size to grow and therefore the probability distributions to change with the block length our primary focus is the situation in which the underlying probabilities are all of the same order and in this regime we show that consistent classification is possible if and only if the alphabet grows subquadratically with the block length we also show that some commonly used statistical tests are suboptimal in that they are consistent only if the alphabet grows sublinearly

leave one out bounds for kernel methods in this article we study leave one out style cross validation bounds for kernel methods the essential element in our analysis is a bound on the parameter estimation stability for regularized kernel formulations using this result we derive bounds on expected leave one out cross validation errors which lead to expected generalization bounds for various kernel algorithms in addition we also obtain variance bounds for leave one out errors we apply our analysis to some classification and regression problems and compare them with previous results

characterizations of matrix and operator valued φ entropies and operator efron stein inequalities we derive new characterizations of the matrix entropy functionals introduced in chen tropp chen tropp 2014 electron j prob 19 1 30 doi 10 1214 ejp v19 2964 1 these characterizations help us to better understand the properties of matrix entropies and are a powerful tool for establishing matrix concentration inequalities for random matrices then we propose an operator valued generalization of matrix entropy functionals and prove the subadditivity under lowner partial ordering our results demonstrate that the subadditivity of operator valued entropies is equivalent to the convexity as an application we derive the operator efron stein inequality 1 http dx doi org 10 1214 ejp v19 2964

approximation of subadditive functions and convergence rates in limiting shape results for a nonnegative subadditive function h on mathbb z d with limiting approximation g x lim n h nx n it is of interest to obtain bounds on the discrepancy between g x and h x typically of order x nu with nu 1 for certain subadditive h x particularly those which are expectations associated with optimal random paths from 0 to x in a somewhat standardized way a more natural and seemingly weaker property can be established every x is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound we show that this convex hull property implies the desired bound for all x applications include rates of convergence in limiting shape results for first passage percolation standard and oriented and longest common subsequences and bounds on the error in the exponential decay approximation to the off axis connectivity function for subcritical bernoulli bond percolation on the integer lattice

probabilistic and worst case analyses of classical problems of combinatorial optimization in euclidean space the classical problems reviewed are the traveling salesman problem minimal spanning tree minimal matching greedy matching minimal triangulation and others each optimization problem is considered for finite sets of points in d and the feature of principal interest is the value of the associated objective function special attention is given to the asymptotic behavior of this value under probabilistic assumptions but both probabilistic and worst case analyses are surveyed

a quantitative central limit theorem for the effective conductance on the discrete torus we study a random conductance problem on a d dimensional discrete torus of size l 0 the conductances are independent identically distributed random variables uniformly bounded from above and below by positive constants the effective conductance al of the network is a random variable depending on l that converges almost surely to the homogenized conductance ahom our main result is a quantitative central limit theorem for this quantity as l in particular we prove there exists some 0 such that dk ld 2 a ahom g l d 2 logd l where dk is the kolmogorov distance and gis a standard normal variable the main achievement of this contribution is the precise asymptotic description of the variance of al 2015 wiley periodicals inc

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

submean variance bound for effective resistance of random electric networks we study a model of random electric networks with bernoulli resistances in the case of the lattice mathbb z 2 we show that the point to point effective resistance between 0 and a vertex v has a variance of order at most rm log v frac 2 3 whereas its expected value is of order log v when v goes to infinity when d 2 expectation and variance are of the same order similar results are obtained in the context of p resistance the proofs rely on a modified poincare inequality due to falik and samorodnitsky 7

random walks on multifractal lattices the properties of random walks on a square lattice with multifractal distributions of site residence probabilities have been explored using computer simulations random walks on such substrates have well defined fractal dimensionalities dw obtained from the dependence of r2 on n where r2 is the mean square displacement from the origin and n is the number of steps in the walk which are larger than two results obtained from these simulations indicate that more than two exponents dw and ds where ds is the fracton or spectral dimensionality are needed to describe the properties of these walks the algorithms used in this work provide a convenient way for generating walks with a fractal dimensionality greater than two and are being used to extend the scope of the diffusion limited aggregation models

normal approximation for a random elliptic equation we consider solutions of an elliptic partial differential equation in mathbb r d with a stationary random conductivity coefficient that is also periodic with period l boundary conditions on a square domain of width l are arranged so that the solution has a macroscopic unit gradient we then consider the average flux that results from this imposed boundary condition it is known that in the limit l rightarrow infty this quantity converges to a deterministic constant almost surely our main result is that the law of this random variable is very close to that of a normal random variable if the domain size l is large we quantify this approximation by an error estimate in total variation the error estimate relies on a second order poincare inequality developed recently by chatterjee

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

relative stability of global errors of nonparametric function estimators this paper presents relative stability properties of various nonparametric density estimators histogram kernel estimates and of regression estimators partitioning kernel and nearest neighbor estimates in density estimation let en denote the l sub 1 error of an estimate calculated from n data whereas in regression estimation the l sub 2 error of the estimate is used sufficient conditions for e sub n e e sub n spl rarr 1 in probability are provided if this limit holds the asymptotic behavior of the random error e sub n can be characterized by its expectation e e sub n and one may apply for example the established rate of convergence results for e en

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

diffusion in one dimensional multifractal porous media we examine the scaling properties of one dimensional random walks on media with multifractal diffusivities which is a simple model for transport in scaling porous media we find both theoretically and numerically that the anomalous scaling exponent of the walk is dw 2 k 1 where k 1 is the scaling exponent of the reciprocal spatially averaged dressed resistance to diffusion since k 1 0 the walk is subdiffusive the walkers are effectively trapped in a hierarchy of barriers the trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion we discuss the implications for transport in porous media

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

theory of earthquakes interevent times applied to financial markets abstract we analyze the probability density function pdf of waiting times between financial loss exceedances the empirical pdfs are fitted with the self excited hawkes conditional poisson process with a long power law memory kernel the hawkes process is the simplest extension of the poisson process that takes into account how past events influence the occurrence of future events by analyzing the empirical data for 15 different financial assets we show that the formalism of the hawkes process used for earthquakes can successfully model the pdf of interevent times between successive market losses

expected length of the longest common subsequence for large alphabets we consider the length l of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k ary alphabet subadditivity arguments yield that e l n converges to a constant k we prove a conjecture of sankoff and mainville from the early 80 s claiming that gamma kappa sqrt k longrightarrow 2 as k longrightarrow infty

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

cumulants of hawkes point processes we derive explicit closed form expressions for the cumulant densities of a multivariate self exciting hawkes point process generalizing a result of hawkes in his earlier work on the covariance density and bartlett spectrum of such processes to do this we represent the hawkes process in terms of a poisson cluster process and show how the cumulant density formulas can be derived by enumerating all possible family trees representing complex interactions between point events we also consider the problem of computing the integrated cumulants characterizing the average measure of correlated activity between events of different types and derive the relevant equations

minimax rates of entropy estimation on large alphabets via best polynomial approximation consider the problem of estimating the shannon entropy of a distribution over k elements from n independent samples we show that the minimax mean square error is within the universal multiplicative constant factors of big k n log k big 2 log 2 k n if n exceeds a constant factor of k log k otherwise there exists no consistent estimator this refines the recent result of valiant and valiant that the minimal sample size for consistent entropy estimation scales according to theta k log k the apparatus of the best polynomial approximation plays a key role in both the construction of optimal estimators and by a duality argument the minimax lower bound

non normal limiting distribution for optimal alignment scores of strings in binary alphabets we consider two independent binary i i d random strings x and y of equal length n and the optimal alignments according to a symmetric scoring functions only we decompose the space of scoring functions into five components two of these components add a part to the optimal score which does not depend on the alignment and which is asymptotically normal we show that when we restrict the number of gaps sufficiently and add them only into one sequence then the alignment score can be decomposed into a part which is normal and has order o sqrt n and a part which is on a smaller order and tends to a tracy widom distribution adding gaps only into one sequence is equivalent to aligning a string with its descendants in case of mutations and deletes for testing relatedness of strings the normal part is irrelevant since it does not depend on the alignment hence it can be safely removed from the test statistic

rates of convergence for quasi additive smooth euclideal functionals and application to combinatorial optimization problems rates of convergence of limit theorems are established for a class of random processes called here quasi additive smooth euclidean functionals examples include the objective functions of the traveling salesman problem the steiner tree problem the minimum spanning tree problem the minimum weight matching problem and a variant of the minimum spanning tree problem with power weighted edges

central limit theorem for first passage percolation time across thin cylinders we prove that first passage percolation times across thin cylinders of the form 0 n hn hn d 1 obey gaussian central limit theorems as long as hn grows slower than n1 d 1 it is an open question as to what is the fastest that hn can grow so that a gaussian clt still holds under the natural but unproven assumption about existence of fluctuation and transversal exponents and strict convexity of the limiting shape in the direction of 1 0 0 we prove that in dimensions 2 and 3 the clt holds all the way up to the height of the unrestricted geodesic we also provide some numerical evidence in support of the conjecture in dimension 2

moment inequalities for functions of independent random variables a general method for obtaining moment inequalities for functions of independent random variables is presented it is a generalization of the entropy method which has been used to derive concentration inequalities for such functions boucheron lugosi and massart ann probab 31 2003 1583 1614 and is based on a generalized tensorization inequality due to latala and oleszkiewicz lecture notes in math 1745 2000 147 168 the new inequalities prove to be a versatile tool in a wide range of applications we illustrate the power of the method by showing how it can be used to effortlessly re derive classical inequalities including rosenthal and kahane khinchine type inequalities for sums of independent random variables moment inequalities for suprema of empirical processes and moment inequalities for rademacher chaos and u statistics some of these corollaries are apparently new in particular we generalize talagrand s exponential inequality for rademacher chaos of order 2 to any order we also discuss applications for other complex functions of independent random variables such as suprema of boolean polynomials which include as special cases subgraph counting problems in random graphs

on the chemical distance for supercritical bernoulli percolation we prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process we improve earlier results of gartner and molchanov and grimmett and marstrand and answer affirmatively a conjecture of kozlov

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

effective permeability of multifractal porous media abstract we show how the real space renormalization group method can be used to calculate analytically the scaling exponents of the effective absolute permeability in multifractal porous media the permeability fields considered are deterministic and random multifractals constructed with multiplicative processes we discuss the implications of the results on the understanding of fluid flow in oil reservoirs

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

rate of convergence of the mean for sub additive ergodic sequences abstract for sub additive ergodic processes x m n with weak dependence we analyze the rate of convergence of e x 0 n n to its limit g we define an exponent given roughly by e x 0 n n g n and assuming existence of a fluctuation exponent that gives var x 0 n n 2 we provide a lower bound for of the form the main requirement is that 1 2 in the case 1 2 and under the assumption var x 0 n o n log n for some 0 we prove c for a dependent constant c these results show in particular that non diffusive fluctuations are associated to non trivial various models including first passage percolation directed polymers the minimum of a branching random walk and bin packing fall into our general framework and the results apply assuming exists in the case of first passage percolation in z d we provide a version of 1 2 without assuming existence of

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

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

localization for random walks among random obstacles in a single euclidean ball place an obstacle with probability 1 p independently at each vertex of mathbb z d and run a simple random walk before hitting one of the obstacles for d ge 2 and p strictly above the critical threshold for site percolation we condition on the environment where the origin is contained in an infinite connected component free of obstacles and we show that for environments with probability tending to one as n rightarrow infty there exists a unique discrete euclidean ball of volume d log 1 p n asymptotically such that the following holds conditioned on survival up to time n we have that at any time t in epsilon n n n for some epsilon n rightarrow n rightarrow infty 0 with probability tending to one the simple random walk is in this ball this work relies on and substantially improves a previous result of the authors on localization in a region of volume poly logarithmic in n for the same problem

about adaptive coding on countable alphabets this paper sheds light on adaptive coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non decreasing hazard rate log concave envelope distributions we prove that the auto censuring ac code is adaptive with respect to the collection of such classes the analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy and on a careful analysis of the performance of the ac coding algorithm the latter relies on nonasymptotic bounds for maxima of samples from discrete distributions with finite and nondecreasing hazard rate

minima in branching random walks given a branching random walk let m n be the minimum position of any member of the n th generation we calculate mathbfem n to within o 1 and prove exponential tail bounds for mathbfp m n mathbfem n x under quite general conditions on the branching random walk in particular together with work by bramson z wahrsch verw gebiete 45 1978 89 108 our results fully characterize the possible behavior of mathbf em n when the branching random walk has bounded branching and step size

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

sharp thresholds for monotone non boolean functions and social choice theory a key fact in the theory of boolean functions f 0 1 n 0 1 is that they often undergo sharp thresholds for example if the function f 0 1 n 0 1 is monotone and symmetric under a transitive action with ep f e and eq f 1 e then q p 0 as n here ep denotes the product probability measure on 0 1 n where each coordinate takes the value 1 independently with probability p the fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well as in social choice in this paper we prove sharp thresholds for monotone functions taking values in an arbitrary finite set we also provide examples of applications of the results to social choice and to random graph problems among the applications is an analog for condorcet s jury theorem and an indeterminacy result for a large class of social choice functions

exact limiting shape for a simplified model of first passage percolation on the plane we derive the limiting shape for the following model of first passage bond percolation on the two dimensional integer lattice the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions the passage times of horizontal bonds are bernoulli distributed while the passage times of vertical bonds are all equal to a deterministic constant to analyze the percolation model we couple it with a one dimensional interacting particle system this particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far away particles we prove a law of large numbers for a tagged particle in this process and the shape result for the percolation is obtained as a corollary

central limit theorems for associated random variables and the percolation model we prove a central limit theorem for families x n n mathbf n in lambda n of associated random variables indexed by subsets lambda n of mathbb z d as n rightarrow infty this is an extension of the newman wright invariance principle for associated stationary sequences x n n geq 1 satisfying sum n operatorname cov x 1 x n infty but with the stationarity property replaced by conditions on the moments of the x s the theorem has applications to the voter model and the percolation model in the latter case it provides an extension of a central limit theorem of the authors 4 by reducing the severity of the moment conditions also we prove a central limit theorem for certain non stationary non associated families of random variables which arise in percolation theory this includes for example a central limit theorem for the number of open clusters contained within the circuit gamma n of mathbb z 2 where gamma n is a sequence of circuits which satisfy a regularity condition and whose interiors overset circ gamma n satisfy overset circ gamma n rightarrow infty as n rightarrow infty

hard and soft multifractal processes abstract we show that multifractal notions encompass a wider variety of phenomena than often believed ranked by increasing highest order of singularities we have geometric then microcanonical and finally canonical multifractals they are respectively localized and calm delocalized and calm and delocalized and wild canonical multifractals may also involve rare violent hard singularities which cause high order statistical moments to diverge we demonstrate this classification in the framework of universal multifractals characterized by three fundamental exponents

on the variance of the optimal alignments score for binary random words and an asymmetric scoring function we investigate the order of the variance of the optimal alignments oa score of two independent iid binary random words having the same length the letters are equiprobable but the scoring function is such that one letter has a larger score than the other in this setting we prove that the order of variance is linear in the common length oas constitute a generalization of longest common subsequences they can be represented as optimal paths in a two dimensional last passage percolation setting with dependent weights

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

random geometric complexes in the thermodynamic regime we consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices in particular we study the betti numbers of these complexes as the number of vertices becomes large obtaining limit theorems for means strong laws concentration inequalities and central limit theorems as opposed to most prior papers treating random complexes the limit with which we work is in the so called thermodynamic regime which includes the percolation threshold in which the complexes become very large and complicated with complex homology characterised by diverging betti numbers the proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of mayer vietoris exact sequences the mayer vietoris arguments are crucial since homology in general and betti numbers in particular are global rather than local phenomena and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality

euclidean networks with a backbone and a limit theorem for minimum spanning caterpillars a caterpillar network or graph g is a tree with the property that removal of the leaf edges of g leaves one with a path here we focus on minimum weight spanning caterpillars where the vertices are points in the euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding euclidean lengths the flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling in particular one can accommodate problems motivated by communications theory such as the last mile problem geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well known beardwood halton and hammersley theorem for the length of the shortest tour through a random sample but the minimal spanning caterpillars fall outside the scope of the theory of subadditive euclidean functionals

thermodynamical approach to the longest common subsequence problem we introduce an interacting particle model in a random media and show that this particle process is equivalent to the longest common subsequence lcs problem of two binary sequences we derive a differential equation which links the mean lcs curve to the average speed of the particles given their density and prove that the average speed of the particles and density converges uniformly on every scale which is somewhat larger than sqrt n

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

an invariance principle for certain dependent sequences let x 1 x 2 cdots be a strictly stationary second order sequence which is associated i e is such that any two coordinatewise nondecreasing functions of the x i s of finite variance are nonnegatively correlated if sum j operatorname cov x 1 x j infty then the partial sum processes w n t defined in the usual way so that w n m n x 1 cdots x m me x 1 sqrt n for m 1 2 cdots converge in distribution on c lbrack 0 t rbrack to a wiener process this result is based on two general theorems concerning associated random variables which are of independent interest

a new method of normal approximation we introduce a new version of stein s method that reduces a large class of normal approximation problems to variance bounding exercises thus making a connection between central limit theorems and concentration of measure unlike skorokhod embeddings the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily as an application we derive a general clt for functions that are obtained as combinations of many local contributions where the definition of local itself depends on the data several examples are given including the solution to a nearest neighbor clt problem posed by p bickel

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

standard deviation of the longest common subsequence let l n be the length of the longest common subsequence of two independent i i d sequences of bernoulli variables of length n we prove that the order of the standard deviation of l n is vn provided the parameter of the bernoulli variables is small enough this validates waterman s conjecture in this situation philos trans r soc lond ser b 344 1994 383 390 the order conjectured by chvatal and sankoff j appl probab 12 1975 306 315 however is different

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

branching ratio approximation for the self exciting hawkes process we introduce a model independent approximation for the branching ratio of hawkes self exciting point processes our estimator requires knowing only the mean and variance of the event count in a sufficiently large time window statistics that are readily obtained from empirical data the method we propose greatly simplifies the estimation of the hawkes branching ratio recently proposed as a proxy for market endogeneity and formerly estimated using numerical likelihood maximization we employ our method to support recent theoretical and experimental results indicating that the best fitting hawkes model to describe s futures price changes is in fact critical now and in the recent past in light of the long memory of financial market activity

a coincidence based test for uniformity given very sparsely sampled discrete data how many independent samples n do we need from a distribution p to decide that p is epsiv distant from uniform in an l1 sense sigmai 1 m p i 1 m epsiv here m is the number of bins on which the distribution is supported and is assumed known a priori somewhat surprisingly we only need n epsiv2 gt m 1 2 to make this decision reliably this condition is both sufficient and necessary the test for uniformity introduced here is based on the number of observed ldquocoincidencesrdquo samples that fall into the same bin the mean and variance of which may be computed explicitly for the uniform distribution and bounded nonparametrically for any distribution that is known to be epsiv distant from uniform some connections to the classical birthday problem are noted