supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

analysis of dispersion of multinomial responses abstract in analogy to analysis of variance for linear models analyses of dispersion for loglinear models for multinomial responses are constructed the analyses which are based on the entropy and concentration measures are used to construct tests of independence and measures of association

on skew brownian motion we consider the stochastic equation x t w t beta l x 0 t where w is a standard wiener process and l x 0 cdot is the local time at zero of the unknown process x there is a unique solution x and it is adapted to the fields of w if beta leq 1 but no solutions exist if beta 1 in the former case setting alpha beta 1 2 the unique solution x is distributed as a skew brownian motion with parameter alpha this is a diffusion obtained from standard wiener process by independently altering the signs of the excursions away from zero each excursion being positive with probability alpha and negative with probability 1 alpha finally we show that skew brownian motion is the weak limit as n rightarrow infty of n 1 2 s lbrack nt rbrack where s n is a random walk with exceptional behavior at the origin

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

the mean velocity of a brownian motion in a random levy potential a brownian motion in a random levy potential v is the informal solution of the stochastic differential equation dx t db t 1 2v x t dt where b is a brownian motion independent of v we generalize some results of kawazu tanaka who considered for v a brownian motion with drift by proving that x t t converges almost surely to a constant the mean velocity which we compute in terms of the levy exponent of v defined by e e mv t e t m

the triangle condition for contact processes on homogeneous trees we complete work of c c wu by showing that for contact processes on homogeneous trees with degree at least 3 the triangle condition is satisfied below the second critical point in particular it holds at the first critical point and therefore at this critical point the contact process has mean field critical exponents

the contact process on trees the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter is varied for small values of a single infection eventually dies out for larger the infection lives forever with positive probability but eventually leaves any finite set the survival probability is a continuous function of and the proof of this is much easier than it is for the contact process on d dimensional integer lattices for still larger the infection converges in distribution to a nontrivial invariant measure for an n ary tree with n large the first of these transitions occurs when 1 n and the second occurs when 1 2 n e n nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n ary trees provided that vertices of degree n are not too rare in particular letting n go to galton watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at 0 the nature of the second phase transition is not yet clear and several problems are mentioned in this regard

a class of logistic regression models for multivariate binary time series abstract a logistic model for multivariate binary time series is proposed first we establish the equivalence between a log linear model for the marginal distribution of a multivariate binary random vector and logistic models for the conditional distributions of each component given the others the logistic formulation is used to describe a markov chain for each series which implies a markov model for the vector of time series a pseudolikelihood estimation procedure is presented the methods are illustrated with data on psychosomatic and psychological diagnoses for families in a health maintenance plan

prediction functions for categorical panel data prediction of categorical responses in panel studies is considered prediction functions based on general conditional log linear models are investigated for statistical properties both from a population perspective and a sampling perspective problems such as existence and uniqueness of optimal prediction functions are addressed and basic properties of measures of prediction quality are examined estimation consistency and asymptotic normality are studied for the proposed parameter estimates and measures of prediction quality

conditional log linear models for analyzing categorical panel data abstract conditional log linear models are developed for panel data and used to predict sequences of categorical responses the class of models considered includes conventional markov models and independence models as well as distance models in which all previous responses and present and past values of covariates are used to predict the current response the approach taken in this article has some advantages over the marginal modeling approach that has become popular for longitudinal studies quality of prediction is measured by using a logarithmic penalty function given a model conditional probabilities of responses consistent with the model are selected to provide the smallest expected penalty this minimum expected penalty provides a measure of the predictive power of a model models are compared through their predictive power as measured by the proportional reduction in expected penalty ways of incorporating the number of parameters of the competing models are discussed this emphasis on predicti

rates of convergence of a transient diffusion in a spectrally negative levy potential we consider a diffusion process x in a random levy potential v which is a solution of the informal stochastic differential equation dx t d t 1 2v x t dt x 0 0 b m independent of v we study the rate of convergence when the diffusion is transient under the assumption that the levy process v does not possess positive jumps we generalize the previous results of hu shi yor for drifted brownian potentials in particular we prove a conjecture of carmona provided that there exists 0 1 such that e e v 1 1 then x t t converges to some nondegenerate distribution these results are in a way analogous to those obtained by kesten kozlov spitzer for the transient random walk in a random environment

zero white noise limit through dirichlet forms with application to diffusions in a random medium we study the zero white noise limit for diffusions in a continuous multidimensional medium given a continuous function on n w we consider diffusions whose drift term is the gradient ofw and whose diffusion coefficient is constant equal to e we describe the asymptotics of the exit time from a domain and of the law of the process when e tends to zero by applying these results to a random self similar mediumw we prove limit theorems for a diffusion in a random medium our theorems agree with results usually proved through the large deviation principle although in our setup this last tool is not available we extend to the multidimensional case properties of diffusions in a random medium already known in one dimension

growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree it is known that the contact process on a homogeneous tree of degree d 1 3 has a weak survival phase in which the infection survives with positive probability but nevertheless eventually vacates every finite subset of the tree it is shown in this paper that in the weak survival phase there exists a spherically symmetric invariant measure whose density decays exponentially at infinity thus confirming a conjecture of liggett the proof is based on a study of the relationships between various thermodynamic parameters and functions associated with the contact process initiated by a single infected site these include 1 the growth profile which determines the exponential rate of growth in space time on the event of survival 2 the exponential rate 3 of decay of the hitting probability function at infinity also studied by the author and 3 the exponential rate q of decay in time t of the probability that the initial infected site is infected at time t it is shown that 83 is a strictly increasing function of the infection rate a in the weak survival phase and that 3 1 fd at the upper critical point a2 demarcating the boundary between the weak and strong survival phases it is also shown that q 1 except at a2 where q 1

the second lowest extremal invariant measure of the contact process ii we continue the investigation of the behavior of the contact process on infinite connected graphs of bounded degree some questions left open by salzano and schonmann 1997 concerning the notions of complete convergence partial convergence and the criterion r s are answered the continuity properties of the survival probability and the recurrence probability are studied these order parameters are found to have a richer behavior than expected with the possibility of the survival probability being discontinuous at or above the threshold for survival a condition which guarantees the continuity of the survival probability above the survival point is introduced and exploited the recurrence probability is shown to always be left continuous above the recurrence point and a necessary and sufficient condition for its right continuity is introduced and exploited it is shown that for homogeneous graphs the survival probability can only be discontinuous at the survival point and the recurrence probability can only be discontinuous at the recurrence point for graphs which are obtained by joining a finite number of severed homogeneous trees by means of a finite number of vertices and edges the survival point the recurrence point and the discontinuity points of the survival and recurrence probabilities are located

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

rates of convergence of diffusions with drifted brownian potentials we are interested in the asymptotic behaviour of a diffusion process with drifted brownian potential the model is a continuous time analogue to the random walk in random environment studied in the classical paper of kesten kozlov and spitzer we not only recover the convergence of the diffusion process which was previously established by kawazu and tanaka but also obtain all the possible convergence rates an interesting feature of our approach is that it shows a clear relationship between drifted brownian potentials and bessel processes

intermediate phase for the contact process on a tree we show that in between the two critical values for the contact process on trees there are infinitely many extremal nontranslation invariant stationary distributions this conclusion is reminiscent of results of grimmett and newman for percolation on the product of a tree with the integers where there is an intermediate phase with infinitely many infinite clusters

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

annealed tail estimates for a brownian motion in a drifted brownian potential we study brownian motion in a drifted brownian potential kawazu and tanaka j math soc japan 49 1997 189 211 exhibited two speed regimes for this process depending on the drift they supplemented these laws of large numbers by central limit theorems which were recently completed by hu shi and yor trans amer math soc 351 1999 3915 3934 using stochastic calculus we studied large deviations ann probab 29 2001 1173 1204 showing among other results that the rate function in the annealed setting that is after averaging over the potential has a flat piece in the ballistic regime in this paper we focus on this subexponential regime proving that the probability of deviating below the almost sure speed has a polynomial rate of decay and computing the exponent in this power law this provides the continuous time analogue of what dembo peres and zeitouni proved for the transient random walk in random environment comm math phys 181 1996 667 683 our method takes a completely different route making use of lamperti s representation together with an iteration scheme

a one dimensional diffusion process in a wiener medium we consider a one dimensional diffusion process in a random medium which is itself generated by a wiener process the asymptotic behaviour is investigated using the self similarity of the medium process the results are analogous to the lattice case

item response theory with estimation of the latent population distribution using spline based densities the purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines a spline based density estimation system provides a flexible alternative to existing procedures that use a normal distribution or a different functional form for the population distribution a simulation study shows that the new procedure is feasible in practice and that when the latent distribution is not well approximated as normal two parameter logistic 2pl item parameter estimates and expected a posteriori scores eaps can be improved over what they would be with the normal model an example with real data compares the new method and the extant empirical histogram approach

the existence of an intermediate phase for the contact process on trees let mathbb t d be a homogeneous tree in which every vertex has d neighbors a new proof is given that the contact process on mathbb t d exhibits two phase transitions when d geq 3 a behavior which distinguishes it from the contact process on mathbb z n this is the first proof which does not involve calculation of bounds on critical values and it is much shorter than the previous proof for the binary tree mathbb t 3 the method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties for which no such result was previously known

the complete convergence theorem of the contact process on trees consider the contact process on a homogeneous tree with degree d 3 denote by c inf p o f t i o 0 the critical value of local survival probability where o is the root of the tree pemantle and durrett and schinazi both conjectured that the complete convergence theorem should hold if c here we answer the conjecture affirmatively furthermore we will show that p o ft i o 0 at c therefore the conclusion of the complete convergence theorem cannot hold at c

discrete schemes for processes in random media we present discrete schemes for processes in random media we prove two results the first one is the convergence of sinai s random walks in random environments to the brox model the second one is the convergence of random walks in media with random gates to a continuous process in a poisson potential the proofs are based on the following idea we consider the discrete media as random potentials for continuous models

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

concavity and estimation on donne des conditions simplifiees pour la consistance et la normalite asymptotique d estimations m deduites par maximisation de moyennes de fonctions concaves aleatoires identiquement distribuees et independantes des applications sont donnees pour l estimation de maximum de vraisemblance

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

a log linear model for ordinal data to characterize differential change among treatments we propose a family of log linear models for ordinal data that contain parameters reflecting change patterns to compare treatments relative to change from baseline under the most general model rates of change can depend not only upon the direction of change but also upon the level of the baseline classification we describe methods for selection of a parsimonious model and for tests of hypotheses concerning treatment differences interpretation of treatment differences in the follow up response profiles within baseline strata employs the concept of stochastic ordering data from two clinical trials illustrate the proposed procedure