on the relation between the standard phase field model and a thermodynamically consistent phase field model abstract a further comparison is made between the standard phase field equations t 2 1 2 g u u t 2 u 1 2 l t and the relevant thermodynamically consistent model of phase transitions proposed by the authors physica d 43 1990 44 62 here we concentrate on the usual case where g 3 and for comparison purposes retain this expression for the analogous nonlinear functions in the latter model it is brought out among other things that the standard model is thermodynamically consistent in the sense of being derivable from a free energy functional however this free energy functional is of a somewhat unusual kind it implies that at constant temperature the energy density varies linearly with the order parameter and the entropy density is a non concave function of the example of a hard sphere system indicates that such behaviour is not impossible but in most other models the energy density and the entropy are both strictly concave in

mdl denoising revisited we refine and extend an earlier minimum description length mdl denoising criterion for wavelet based denoising we start by showing that the denoising problem can be reformulated as a clustering problem where the goal is to obtain separate clusters for informative and noninformative wavelet coefficients respectively this suggests two refinements adding a code length for the model index and extending the model in order to account for subband dependent coefficient distributions a third refinement is the derivation of soft thresholding inspired by predictive universal coding with weighted mixtures we propose a practical method incorporating all three refinements which is shown to achieve good performance and robustness in denoising both artificial and natural signals

lasalle type theorems for stochastic differential delay equations the main aim of this paper is to establish the lasalle type asymptotic convergence theorems for the solutions of stochastic differential delay equations these stochastic versions are then applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations several examples are also given for illustration

learning high dimensional markov forest distributions analysis of error rates the problem of learning forest structured discrete graphical models from i i d samples is considered an algorithm based on pruning of the chow liu tree through adaptive thresholding is proposed it is shown that this algorithm is both structurally consistent and risk consistent and the error probability of structure learning decays faster than any polynomial in the number of samples under fixed model size for the high dimensional scenario where the size of the model d and the number of edges k scale with the number of samples n sufficient conditions on n d k are given for the algorithm to satisfy structural and risk consistencies in addition the extremal structures for learning are identified we prove that the independent resp tree model is the hardest resp easiest to learn using the proposed algorithm in terms of error rates for structure learning

asymptotically tight bounds on the depth of estimated context trees we study the maximum markov model order that can be estimated for an individual input sequence x of length n over a finite alphabet of size alpha for popular estimators of tree models where the estimated order is determined by the depth of a context tree estimate and in the special case of plain markov models where the tree is constrained to be perfect with all leaves at the same depth first we consider penalized maximum likelihood estimators where a context tree hat t is obtained by minimizing a cost of the form log hat p t x f n s t where hat p t x is the ml of x under a model with context tree t s t is the set of leaves of t and f n is an increasing penalization function of n the popular bic estimator is a special case with f n frac alpha 1 2 log n for plain markov models a simple argument yields a known upper bound overline k n o log n on the maximum order that can be estimated for x and we show that in fact this simple bound is not far from tight for general context trees we derive an asymptotic upper bound n 1 2 o 1 on the estimated depth and we exhibit explicit input sequences that asymptotically attain the bound up to a multiplicative constant factor we show that a similar upper bound applies also to mdl estimators based on the kt probability assignment and moreover the same example sequences asymptotically approach the upper bound also in this case

numerical solutions of stochastic differential delay equations under the generalized khasminskii type conditions the classical existence and uniqueness theorem of the solution to a stochastic differential delay equation sdde requires the local lipschitz condition and the linear growth condition see e g 11 12 and 20 the numerical solutions under these conditions have also been discussed intensively see e g 4 10 13 16 17 18 21 22 and 24 recently mao and rassias 14 and 15 established the generalized khasminskii type existence and uniqueness theorems for sddes where the linear growth condition is no longer imposed these generalized khasminskii type theorems cover a wide class of highly nonlinear sddes but these nonlinear sddes do not have explicit solutions whence numerical solutions are required in practice however there is so far little numerical theory on sddes under these generalized khasminskii type conditions the key aim of this paper is to close this gap

data driven calibration of penalties for least squares regression penalization procedures often suffer from their dependence on multiplying factors whose optimal values are either unknown or hard to estimate from data we propose a completely data driven calibration algorithm for these parameters in the least squares regression framework without assuming a particular shape for the penalty our algorithm relies on the concept of minimal penalty recently introduced by birge and massart 2007 in the context of penalized least squares for gaussian homoscedastic regression on the positive side the minimal penalty can be evaluated from the data themselves leading to a data driven estimation of an optimal penalty which can be used in practice on the negative side their approach heavily relies on the homoscedastic gaussian nature of their stochastic framework the purpose of this paper is twofold stating a more general heuristics for designing a data driven penalty the slope heuristics and proving that it works for penalized least squares regression with a random design even for heteroscedastic non gaussian data for technical reasons some exact mathematical results will be proved only for regressogram bin width selection this is at least a first step towards further results since the approach and the method that we use are indeed general

redundancy of the context tree weighting method on renewal and markov renewal processes the context tree weighting method ctw is shown to be almost adaptive on the classes of renewal and markov renewal processes up to logarithmic factor ctw achieves the minimax pointwise redundancy described by i csiszaacuter and p shields in ieee trans inf theory vol 42 no 6 pp 2065 2072 nov 1996 this result not only complements previous results on the adaptivity of the context tree weighting method on the relatively small class of all finite context tree sources which encompasses the class of all finite order markov sources it shows that almost minimax redundancy can be achieved on massive classes of sources classes that cannot be smoothly parameterized by subsets of finite dimensional spaces moreover it shows that almost adaptive compression can be achieved in a computationally efficient way on those massive classes while previous adaptivity results for ctw could rely on the fact that any markov source is a finite context tree source this is no longer the case for renewal sources in order to prove almost adaptivity of ctw over renewal sources it is necessary to establish that ctw carefully balances estimation error and approximation error

application of kolmogorov complexity and universal codes to identity testing and nonparametric testing of serial independence for time series we show that kolmogorov complexity and such its estimators as universal codes or data compression methods can be applied for hypotheses testing in a framework of classical mathematical statistics the methods for identity testing and nonparametric testing of serial independence for time series are suggested

universal tests for memory words the main result is a universal pointwise test that when presented with a set of words s on a finite or countable alphabet x that purports to be a set of memory words for a stationary process will eventually almost surely return the value yes precisely when all positive probability words in s are memory words for example if s consists of all of the single letters in x then the test will eventually say yes if and only if the process is a markov chain various further positive and negative results of this type are also given

solutions to a penrose fife model of phase field type abstract in a recent paper sprekels and zheng study global existence of smooth solutions to a model of phase field type for the kinetics of phase transitions which was proposed by penrose and fife in this paper we study this model with constraint and show existence of solutions to that problem

the conserved penrose fife system with fourier heat flux law abstract a system of phase field equations of penrose fife type governing the dynamics of phase transitions with a conserved order parameter is considered as in the original model the heat flux is assumed to be given by the fourier law existence of weak solutions is proved for the related initial neumann boundary value problem

consistent estimation of the basic neighborhood of markov random fields for markov random fields on z d with finite state space we address the statistical estimation of the basic neighborhood the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given a modification of the bayesian information criterion replacing likelihood by pseudo likelihood is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions the estimated basic neighborhood equals the true one eventually almost surely not assuming any prior bound on the size of the latter stationarity of the markov field is not required and phase transition does not affect the results

the conserved penrose fife system with temperature dependent memory abstract a nonlinear parabolic system of penrose fife type with a singular evolution term arising from modelling dynamic phenomena of the nonisothermal diffusive phase separation is studied here we consider the evolution of a material in which the heat flux is a superposition of two different contributions one part is proportional to the spacial gradient of the inverse of the absolute temperature while the other agrees with the gurtin pipkin law introduced in the theory of materials with thermal memory the phase transition here is described through the evolution of the conserved order parameter which may represent the density or concentration of some substance it is shown that an initial boundary value problem for the resulting state equations has a unique solution

markov chain order estimation with conditional mutual information abstract we introduce the conditional mutual information cmi for the estimation of the markov chain order for a markov chain of k symbols we define cmi of order m i c m as the mutual information of two variables in the chain being m time steps apart conditioning on the intermediate variables of the chain we find approximate analytic significance limits based on the estimation bias of cmi and develop a randomization significance test of i c m where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence the significance test is applied for increasing m and the markov chain order is estimated by the last order for which the null hypothesis is rejected we present the appropriateness of cmi testing on monte carlo simulations and compare it to the akaike and bayesian information criteria the maximal fluctuation method peres shields estimator and a likelihood ratio test for increasing orders using divergence the order criterion of cmi testing turns out to be superior for orders larger than one but its effectiveness for large orders depends on data availability in view of the results from the simulations we interpret the estimated orders by the cmi testing and the other criteria on genes and intergenic regions of dna chains

bic context tree estimation for stationary ergodic processes context trees of arbitrary stationary ergodic processes with finite alphabets are considered such a process is not necessarily a markov chain so the context tree may be of infinite depth calculated from a sample of size n the bayesian information criterion bic is shown to provide a strongly consistent estimator of the context tree of the process via minimization over hypothetical context trees without any restriction on the hypothetical context trees strong consistency means that the estimated context tree recovers the true one up to a level k eventually almost surely as n tends to infinity under some conditions on the process it is shown that the recovery level k can grow with n at a specific rate determined by the distribution of the process thus the bic estimator can recover the true context tree to larger and larger depths the results include for the special case of k being an arbitrary constant that the strong consistency is satisfied without any assumption on the stationary ergodic process which itself improves the existing results where either the true context tree was assumed to be of finite depth or the depth of the hypothetical context trees was bounded by o log n

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

testing the order of a model this paper deals with order identification for nested models in the i i d framework we study the asymptotic efficiency of two generalized likelihood ratio tests of the order they are based on two estimators which are proved to be strongly consistent a version of stein s lemma yields an optimal underestimation error exponent the lemma also implies that the overestimation error exponent is necessarily trivial our tests admit non trivial underestimation error exponents the optimal underestimation error exponent is achieved in some situations the overestimation error can decay exponentially fast with respect to a positive power of the number of observations these results are proved under mild assumptions by relating the underestimation resp overestimation error to large resp moderate deviations of the log likelihood process in particular it is not necessary that the classical cramer condition be satisfied namely the log densities are not required to admit every exponential moment three benchmark examples with specific difficulties location mixture of normal distributions abrupt changes and various regressions are detailed so as to illustrate the generality of our results

a mathematical model of dynamics of non isothermal phase separation abstract a mathematical model of thermally activated phase separation in binary systems is proposed the construction is based on the landau ginzburg theory of phase transitions and non equilibrium thermodynamics the proposed system of differential equations represents an extension of the cahn hilliard model to the non isothermal situation the model is shown to be consistent with the first principles of thermodynamics the stability solutions is considered and for the one dimensional case numerical results are presented

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

deinterleaving finite memory processes via penalized maximum likelihood we study the problem of deinterleaving a set of finite memory markov processes over disjoint finite alphabets which have been randomly interleaved by a finite memory switch the deinterleaver has access to a sample of the resulting interleaved process but no knowledge of the number or structure of the component markov processes or of the switch we study conditions for uniqueness of the interleaved representation of a process showing that certain switch configurations as well as memoryless component processes can cause ambiguities in the representation we show that a deinterleaving scheme based on minimizing a penalized maximum likelihood cost function is strongly consistent in the sense of reconstructing almost surely as the observed sequence length tends to infinity a set of component and switch markov processes compatible with the original interleaved process furthermore under certain conditions on the structure of the switch including the special case of a memoryless switch we show that the scheme recovers all possible interleaved representations of the original process experimental results are presented demonstrating that the proposed scheme performs well in practice even for relatively short input samples

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

space efficient representation of truncated suffix trees with applications to markov order estimation suffix trees st are useful in many text processing applications for example to determine the number of occurrences of patterns of arbitrary length in an input string x if the length n of x is large the memory required to represent the st may become a practical performance bottleneck this problem can be alleviated in cases where a nontrivial upper bound is known on the lengths of the patterns of interest by using a truncated st tst however conventional tst implementations still require n bits of memory since they store x we describe a new tst representation that avoids this limitation by storing all the information necessary to reconstruct the tst edge labels in a string y that is often much shorter than x we apply tsts to the implementation of markov order estimators where an upper bound k n on the estimated order can be derived or it is imposed for consistency for example the new representation allows for estimator implementations with sublinear space complexity in some cases of interest in other cases we show experimentally that even when the new representation does not have an asymptotic advantage it still achieves very significant memory savings in practice

on strong consistency of model selection in classification this paper considers model selection in classification in many applications such as pattern recognition probabilistic inference using a bayesian network prediction of the next in a sequence based on a markov chain the conditional probability p y y x x of class yisiny given attribute value xisinx is utilized by model we mean the equivalence relation in x for x x isinxx x harrp y y x x p y y x x forall yisiny by classification we mean the number of such equivalence classes is finite we estimate the model from n samples zn xi yi i 1 n isin xtimesy n using information criteria in the form empirical entropy h plus penalty term k 2 dn the model such that h k 2 dn is minimized is the estimated model where k is the number of independent parameters in the model and dn n 1 infin is a real nonnegative sequence such that lim supndn n 0 for autoregressive processes although the definitions of h and k are different it is known that the estimated model almost surely coincides with the true model as nrarrinfin if dn n 1 infin 2loglogn n 1 infin and that it does not if dn n 1 infin 2loglogn n 1 infin hannan and quinn the problem whether the same property is true for classification was open this paper solves the problem in the affirmative

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

global smooth solutions to a thermodynamically consistent model of phase field type in higher space dimensions abstract in a recent paper penrose and fife proposed a thermodynamically consistent model of phase field type for phase transition phenomena such as liquid solid phase transitions the model is based on the observation that the second law of thermodynamics postulates that the entropy functional cannot decrease along solution paths it turns out that the resulting field equations form a system of partial differential equations which is considerably more difficult than the phase field equations studied by caginalp and others for one space dimension global existence uniqueness and asymptotic behaviour of smooth solutions of the initial boundary value problem with neumann boundary data have been obtained in a recent paper by zheng in this paper we prove global existence and uniqueness in three space dimensions for the initial boundary value problem with a third boundary condition

optimal error exponents in hidden markov models order estimation we consider the estimation of the number of hidden states the order of a discrete time finite alphabet hidden markov model hmm the estimators we investigate are related to code based order estimators penalized maximum likelihood ml estimators and penalized versions of the mixture estimator introduced by liu and narayan 1994 we prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works we prove a version of stein s lemma for hmm order estimation and derive an upper bound on underestimation exponents then we prove that this upper bound can be achieved by the penalized ml estimator and by the penalized mixture estimator the proof of the latter result gets around the elusive nature of the ml in hmm by resorting to large deviation techniques for empirical processes finally we prove that for any consistent hmm order estimator for most hmm the overestimation exponent is null

strong convergence of split step backward euler method for stochastic differential equations with non smooth drift in this paper we are concerned with the numerical approximation of stochastic differential equations with discontinuous nondifferentiable drifts we show that under one sided lipschitz and general growth conditions on the drift and global lipschitz condition on the diffusion a variant of the implicit euler method known as the split step backward euler ssbe method converges with strong order of one half to the true solution our analysis relies on the framework developed in d j higham x mao and a m stuart strong convergence of euler type methods for nonlinear stochastic differential equations siam journal on numerical analysis 40 2002 1041 1063 and exploits the relationship which exists between explicit and implicit euler methods to establish the convergence rate results

estimation of the number of communities in the stochastic block model in this article we introduce an estimator for the number of communities in the stochastic block model sbm based on the maximization of a penalized version of the so called krichevsky trofimov mixture distribution we prove its eventual almost sure convergence to the underlying number of communities without assuming a known upper bound on that quantity our results apply to both the dense and the sparse regimes to our knowledge this is the first consistency result for the estimation of the number of communities in the sbm in the unbounded case that is when the number of communities is allowed to grow with the same size

consistent order estimation and minimal penalties consider an i i d sequence of random variables whose distribution f lies in one of the nested families of models mq q 1 the smallest index q such that mq contains f is called the model order the aim of this paper is to explore the consistency properties of penalized likelihood model order estimators such as bayesian information criterion we show in a general setting that the minimal strongly consistent penalty is of order q loglogn where q is a dimensional quantity in contrast to previous work an a priori upper bound on the model order is not assumed the results rely on a sharp characterization of the pathwise fluctuations of the generalized likelihood ratio statistic under entropy assumptions on the model classes our results are applied to the geometrically complex problem of location mixture order estimation which is widely used but poorly understood

error exponents for ar order testing this paper is concerned with error exponents in testing problems raised by autoregressive ar modeling the tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving average arma modeling estimation in several related problems such as markov order or hidden markov model order estimation optimal error exponents have been determined thanks to large deviations theory ar order testing is specially challenging since the natural tests rely on quadratic forms of gaussian processes in sharp contrast with empirical measures of markov chains the large deviation principles ldps satisfied by gaussian quadratic forms do not always admit an information theoretic representation despite this impediment we prove the existence of nontrivial error exponents for gaussian ar order testing and furthermore we exhibit situations where the exponents are optimal these results are obtained by showing that the log likelihood process indexed by ar models of a given order satisfy an ldp upper bound with a weakened information theoretic representation

consistency of the unlimited bic context tree estimator the bayesian information criterion bic and krichevsky trofimov kt version of minimum description length mdl principle are popular in the study of model selection for order estimation of markov chains both are known to be strongly consistent when there is an upper bound on the order in the unbounded case the bic is also known to be consistent but the kt estimator is consistent only with a bound o log n on the order for context trees a flexible generalization of markov models widely used in data processing the problem is more complicated both in theory and practice given the substantially higher number of possible candidate models imre csiszar and zsolt talata proved the consistency of bic and kt when the hypothetical tree depths are allowed to grow as o log n this correspondence proves that such a restriction is not necessary for finite context sources the bic context tree estimator is strongly consistent even if there is no constraint at all on the size of the chosen tree moreover an algorithm computing the tree minimizing the bic criterion among all context trees in linear time is provided

convergence rate of numerical solutions to sfdes with jumps abstract in this paper we are interested in numerical solutions of stochastic functional differential equations with jumps under a global lipschitz condition we show that the p th moment convergence of euler maruyama numerical solutions to stochastic functional differential equations with jumps has order 1 p for any p 2 this is significantly different from the case of stochastic functional differential equations without jumps where the order is 1 2 for any p 2 it is therefore best to use the mean square convergence for stochastic functional differential equations with jumps moreover under a local lipschitz condition we reveal that the order of mean square convergence is close to 1 2 provided that local lipschitz constants valid on balls of radius j do not grow faster than log j

performance analysis of grammar based codes revisited the compression performance of grammar based codes is revisited from a new perspective previously the compression performance of grammar based codes was evaluated against that of the best arithmetic coding algorithm with finite contexts in this correspondence we first define semifinite state sources and finite order semi markov sources based on the definitions of semifinite state sources and finite order semi markov sources and the idea of run length encoding rle we then extend traditional rle algorithms to context based rle algorithms rle algorithms with k contexts and rle algorithms of order k where k is a nonnegative integer for each individual sequence x let r sup sub sr k x and r sup sub sr k x be the best compression rate given by rle algorithms with k contexts and by rle algorithms of order k respectively it is proved that for any x r sup sub sr k is no greater than the best compression rate among all arithmetic coding algorithms with k contexts furthermore it is shown that there exist stationary ergodic semi markov sources for which the best rle algorithms without any context outperform the best arithmetic coding algorithms with any finite number of contexts finally we show that the worst case redundancies of grammar based codes against r sup sub sr k x and r sup sub sr k x among all length n individual sequences x from a finite alphabet are upper bounded by d sub 1 loglogn logn and d sub 2 loglogn logn respectively where d sub 1 and d sub 2 are constants this redundancy result is stronger than all previous corresponding results

consistent estimator for basis selection based on a proxy of the kullback leibler distance abstract given a random sample from a continuous and positive density f the logistic transformation is applied and a log density estimate is provided by using basis functions approach the number of basis functions acts as the smoothing parameter and it is estimated by minimizing a penalized proxy of the kullback leibler distance which includes as particular cases aic and bic criteria we prove that this estimator is consistent

twice universal simulation of markov sources and individual sequences the problem of universal simulation given a training sequence is studied both in a stochastic setting and for individual sequences in the stochastic setting the training sequence is assumed to be emitted by a markov source of unknown order extending previous work where the order is assumed known and leading to the notion of twice universal simulation a simulation scheme which partitions the set of sequences of a given length into classes is proposed for this setting and shown to be asymptotically optimal this partition extends the notion of type classes to the twice universal setting in the individual sequence scenario the same simulation scheme is shown to generate sequences which are statistically similar in a strong sense to the training sequence for statistics of any order while essentially maximizing the uncertainty on the output

prediction for discrete time series let x n be a stationary and ergodic time series taking values from a finite or countably infinite set assume that the distribution of the process is otherwise unknown we propose a sequence of stopping times n along which we will be able to estimate the conditional probability p x x0 from data segment x0 in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily markovian processes if the stationary and ergodic process turns out to be finitarily markovian among others all stationary and ergodic markov chains are included in this class then almost surely if the stationary and ergodic process turns out to possess finite entropy rate then n is upperbounded by a polynomial eventually almost surely

non linear system for non isothermal diffusive phase separation abstract an evolution system which extends the cahn hilliard model to non isothermal situations subject to convex constraints for solutions is studied the problem can be considered as representing a simple model of non isothermal phase separation driven by diffusive mechanisms a variational formulation of the problem is introduced using t dependent subdifferential operator techniques in hilbert spaces existence and uniqueness of weak solutions are proved

the consistency of the bic markov order estimator the bayesian information criterion bic estimates the order of a markov chain with finite alphabet a from observation of a sample path x 1 x 2 x n as that value k k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term a k a 1 2 log n we show that k equals the correct order of the chain eventually almost surely as n oo thereby strengthening earlier consistency results that assumed an apriori bound on the order a key tool is a strong ratio typicality result for markov sample paths we also show that the bayesian estimator or minimum description length estimator of which the bic estimator is regarded as an approximation fails to be consistent for the uniformly distributed i i d process

weak solutions to a penrose fife model with fourier law for the temperature abstract we prove global existence in time of weak solutions to the penrose fife model when the heat flux is given by the fourier law and the homogeneous free energy density by the classical double well potential the main feature of these weak solutions is the almost everywhere positivity of the absolute temperature throughout time evolution

infinite horizon optimal control of forward backward stochastic differential equations with delay we consider a problem of optimal control of an infinite horizon system governed by forward backward stochastic differential equations with delay sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived we illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay

stochastic differential delay equation moment stability and application to hematopoietic stem cell regulation system we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise the results established here are applied to examining the local stability of the hematopoietic stem cell hsc regulation system in the presence of noise the stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system however the stability of the second moment is altered by the perturbation we obtain using laplace transform techniques necessary and sufficient conditions for the second moment to be bounded in applying the results to the hsc system we find that the system stability is sensitive to perturbations in the stem cell differentiation and death rates but insensitive to perturbations in the proliferation rate

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

numerical solutions of stochastic differential delay equations under local lipschitz condition in this paper a variant of the euler maruyama method is used to define the numerical solutions for stochastic differential delay equations sddes with variable delay the key aim is to show that the numerical solutions will converge to the true solutions of the sddes under the local lipschitz condition

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

estimating the lengths of memory words for a stationary stochastic process xn with values in some set a a finite word w isin ak is called a memory word if the conditional probability of x 0 given the past is constant on the cylinder set defined by x k 1 w it is a called a minimal memory word if no proper suffix of w is also a memory word for example in a k step markov processes all words of length k are memory words but not necessarily minimal we consider the problem of determining the lengths of the longest minimal memory words and the shortest memory words of an unknown process xn based on sequentially observing the outputs of a single sample xi1 xi2 xin we will give a universal estimator which converges almost surely to the length of the longest minimal memory word and show that no such universal estimator exists for the length of the shortest memory word the alphabet a may be finite or countable

hidden markov models with applications in cell adhesion experiments estimation of the number of hidden states is challenging in hidden markov models motivated by the analysis of a specific type of cell adhesion experiments a new framework based on a hidden markov model and double penalized order selection is proposed the order selection procedure is shown to be consistent in estimating the number of states a modified expectation maximization algorithm is introduced to efficiently estimate parameters in the model simulations show that the proposed framework outperforms existing methods applications of the proposed methodology to real data demonstrate the accuracy of estimating receptor ligand bond lifetimes and waiting times which are essential in kinetic parameter estimation supplementary materials for this article are available online

large scale typicality of markov sample paths and consistency of mdl order estimators for markov chains of arbitrary order with finite alphabet a almost sure sense limit theorems are proved on relative frequencies of k blocks and of symbols preceded by a given k block when k is permitted to grow as the sample size n grows as an application the consistency of two kinds of minimum description length mdl markov order estimators is proved with upper bound o log n respectively spl alpha log n with spl alpha 1 log a on the permissible value of the estimated order it was shown by csiszar and shields see ann statist vol 28 p 1601 1619 2000 that in the absence of any bound or with bound spl alpha log n with large spl alpha consistency fails

weak solutions to the penrose fife phase field model for a class of admissible heat flux laws abstract this paper is concerned with a thermodynamically consistent model for diffusive phase transitions proposed by penrose and fife the model has recently received a good deal of attention and the related initial boundary value problems have been investigated from the mathematical point of view in the case where the order parameter is not conserved the common working assumption for the heat flow was that such a flux is proportional to the gradient of the inverse absolute temperature this position seems to be helpful for the analysis due to the coupling term in the phase field equation which depends right on the inverse temperature here we prove the existence of weak solutions in a wide setting of constitutive laws for the heat flux different from the previous ones and considerably more significant owing to their behaviour for high temperatures we also discuss the limiting situation of zero interfacial energy for the order parameter

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

context tree estimation for not necessarily finite memory processes via bic and mdl the concept of context tree usually defined for finite memory processes is extended to arbitrary stationary ergodic processes with finite alphabet these context trees are not necessarily complete and may be of infinite depth the familiar bayesian information criterion bic and minimum description length mdl principles are shown to provide strongly consistent estimators of the context tree via optimization of a criterion for hypothetical context trees of finite depth allowed to grow with the sample size n as o logn algorithms are provided to compute these estimators in o n time and to compute them on line for all i les n in o nlogn time

order estimation of markov chains estimators spl chi sub n x sub 0 x sub 1 x sub n are described which when applied to an unknown stationary process taking values from a countable alphabet spl chi converge almost surely to k in case the process is a kth order markov chain and to infinity otherwise

non isothermal phase transition models with neumann boundary conditions abstract we study the systems of nonlinear parabolic pdes for penrose fife models of non isothermal phase transitions both models with conserved and non conserved order parameters are studied with neumann boundary conditions imposed on the systems in multi dimensional spatial domain

compression based methods for nonparametric prediction and estimation of some characteristics of time series we address the problem of online prediction for time series we show that any universal code or a universal data compressor can be used as a basis for constructing asymptotically optimal methods for this problem for a certain class of stationary and ergodic processes

on the minimal penalty for markov order estimation we show that large scale typicality of markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale as a consequence the penalized likelihood markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n our method of proof using techniques from empirical process theory does not rely on the explicit expression for the maximum likelihood estimator in the markov case and could therefore be applicable in other settings