some limit theorems for the empirical process indexed by functions let n 1 be a sequence of classes of real valued measurable functions defined on a probability space s p under weak metric entropy conditions on n 1 and under growth conditions on we show that there are non zero numerical constantsc1 andc2 such that where n is a non decreasing function ofn related to the metric entropy of a few applications of this general result are considered we obtain a s rates of uniform convergence for the empirical process indexed by intervals as well as a s rates of uniform convergence for the empirical characteristic function over expanding intervals

on strong invariance principles under dependence assumptions on obtient des principes d invariance forts avec un ordre d approximation o t 1 2 k pour des suites de variables aleatoires dependantes

controllability of impulsive neutral stochastic functional differential inclusions with infinite delay this paper deals with the controllability of a class of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space sufficient conditions for the controllability are derived with the help of the fixed point theorem for discontinuous multi valued operators due to dhage an example is provided to illustrate the obtained theory

asymptotic behaviour of a class of stochastic approximation procedures letb be a real banach space considering stochastic approximation procedures of robbins monro and kiefer wolfowitz type one is led to recursion formulas of the form u k 1 u k k 1 gamma k u k k rho v k k eta t k and positive real numbers uk vk tkb valued random variables k random linear operatorsb b where in addition to some further assumptions k almost surely andtk t say almost surely letting y k n k 1 n n n be an array ofb valued random variables such that the partial sums sum limits k 1 m v k and sum limits k 1 m y k n do not differ too much uniformly inm 1 n we shall investigate the distance of u m m 1 n and u n m m 1 n where u k 1 left n right u k left n right k 1 gamma u k left n right k rho y k n k eta t fork 1 n 1 the proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence un

time scales of spike train correlation for neural oscillators with common drive we examine the effect of the phase resetting curve on the transfer of correlated input signals into correlated output spikes in a class of neural models receiving noisy superthreshold stimulation we use linear response theory to approximate the spike correlation coefficient in terms of moments of the associated exit time problem and contrast the results for type i vs type ii models and across the different time scales over which spike correlations can be assessed we find that on long time scales type i oscillators transfer correlations much more efficiently than type ii oscillators on short time scales this trend reverses with the relative efficiency switching at a time scale that depends on the mean and standard deviation of input currents this switch occurs over time scales that could be exploited by downstream circuits

diffusion approximation for slow motion in fully coupled averaging in systems which combine fast and slow motions it is usually impossible to study directly corresponding two scale equations and the averaging principle suggests to approximate the slow motion by averaging in fast variables we consider the averaging setup when both fast and slow motions are diffusion processes depending on each other fully coupled and show that there exists a diffusion process which approximates the slow motion in the l 2 sense much better than the averaged motion prescribed by the averaging principle

an almost sure invariance principle for triangular arrays of banach space valued random variables we give a simpler proof of the probability invariance principle for triangular arrays of independent identically distributed random variables with values in a separable banach space recently proved by de acosta 1 and improve this result to an almost sure invariance principle

the central limit theorem for stochastic processes the main result is that the necessary and sufficient conditions for the central limit theorem for centered second order processes given by gine and zinn 6 can be obtained without any basic measurability condition furthermore we extend some of their results

lp strong convergence of the averaging principle for slow fast spdes with jumps abstract the averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component this paper concerns the l p strong convergence of the averaging principle for two time scales stochastic partial differential equations spdes driven by wiener processes and poisson jumps to achieve this a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation where the dissipative conditions are needed furthermore it is shown that under suitable assumptions the slow component l p strongly converges to the solution of the averaged equation the rate of the convergence is also obtained as a byproduct

limit theorems for functionals of mixing processes with applications to u statistics and dimension estimation in this paper we develop a general approach for investigating the asymptotic distribution of functional xn f zn k k z of absolutely regular stochastic processes zn n z such functional occur naturally as orbits of chaotic dynamical systems and thus our results can be used to study probabilistic aspects of dynamical systems we first prove some moment inequalities that are analogous to those for mixing sequences with their help several limit theorems can be proved in a rather straightforward manner we illustrate this by re proving a central limit theorem of ibragimov and linnik then we apply our techniques to u statistics matrix equation with symmetric kernel h r r r we prove a law of large numbers extending results of aaronson burton dehling gilat hill and weiss for absolutely regular processes we also prove a central limit theorem under a different set of conditions than the known results of denker and keller as our main application we establish an invariance principle for u processes un h h indexed by some class of functions we finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system

limit distributions of u statistics resampled by symmetric stable laws if y i and v i are independent random sequences such thaty i are i i d random variables belonging to the normal domain of attraction of a symmetric stable law 0 2 andv i are i i d random variables then the limit distributions of u statistics n 1 alpha sum limits 1 leqq i t ldots i d leqq n y i 1 ldots y i d f v i 1 ldots v i d coincide with the probability laws of multiple stochastic integralsx d f f t 1 t d dx t d with respect to a symmetric stable processx t

strong invariance principles for mixing random fields given a random field z q indexed by q tuples of positive integers and satisfying a strong mixing condition we study the approximation of the partial sum field sn n z q by brownian sheet setting g alpha n 1 n q in z q n k underset raise0 3em hbox smash scriptscriptstyle geqslant mathop pi limits 1 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant q i pm k n i alpha k 1 q for 0 0 we also construct an example showing that in a somewhat larger similar type domain the above approximation is generally impossible even with 0

relationships between donsker classes and sobolev spaces let h be a separable real hubert space let x be a second countable topological space and x p be a probability space where is the borel sets let t h c b x be linear and continuous then the image of the unit ball of h is a donsker class so if k 1 l 2 then the unit ball of the sobolev space f in l 2 mathbb r n smallint hat fk 2 tfrac 1 2 infty is a donsker class for any p for most other k it is not a donsker class for any p with a bounded density

maxima of asymptotically gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices several classical results on boundary crossing probabilities of brownian motion and random walks are extended to asymptotically gaussian random fields which include sums of i i d random variables with multidimensional indices multivariate empirical processes and scan statistics in change point and signal detection as special cases some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain clumps around high level crossings we also discuss how these results are related to the poisson clumping heuristic and tube formulas of gaussian random fields and describe their applications to laws of the iterated logarithm in the form of the kolmogorov erdos feller integral tests

vector quantile regression an optimal transport approach we propose a notion of conditional vector quantile function and a vector quantile regression a conditional vector quantile function cvqf of a random vector y taking values in d given covariates z z taking values in k is a map u qy z u z which is monotone in the sense of being a gradient of a convex function and such that given that vector u follows a reference non atomic distribution fu for instance uniform distribution on a unit cube in d the random vector qy z u z has the distribution of y conditional on z z moreover we have a strong representation y qy z u z almost surely for some version of u the vector quantile regression vqr is a linear model for cvqf of y given z under correct specification the notion produces strong representation y u f z for f z denoting a known set of transformations of z where u u f z is a monotone map the gradient of a convex function and the quantile regression coefficients u u have the interpretations analogous to that of the standard scalar quantile regression as f z becomes a richer class of transformations of z the model becomes nonparametric as in series modelling a key property of vqr is the embedding of the classical monge kantorovich s optimal transportation problem at its core as a special case in the classical case where y is scalar vqr reduces to a version of the classical qr and cvqf reduces to the scalar conditional quantile function several applications to diverse problems such as multiple engel curve estimation and measurement of financial risk are considered

l2 regularity for parabolic partial integrodifferential equations with delay in the highest order derivatives abstract a partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay by means of abstract regularity results global existence and uniqueness of a strict solution are obtained moreover a characterization of the infinitesimal generator of the solution operator is given

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

some implications of renormalization group theoretical ideas to statistics renormalization group rg theory and statistical data analysis are obviously closely related reductive rg can provide a natural framework to understand asymptotic estimate problems some observations relevant to this problem and an example of a data mining algorithm inspired by the consideration are exhibited

class ii neurons display a higher degree of stochastic synchronization than class i neurons we describe the relationship between the shape of the phase resetting curve prc and the degree of stochastic synchronization observed between a pair of uncoupled general oscillators receiving partially correlated poisson inputs in addition to inputs from independent sources we use perturbation methods to derive an expression relating the shape of the prc to the probability density function pdf of the phase difference between the oscillators we compute various measures of the degree of synchrony and cross correlation from the pdf s and use the same to compare and contrast differently shaped prcs with respect to their ability to undergo stochastic synchronization since the shape of the prc depends on underlying dynamical details of the oscillator system we utilize the results obtained from the analysis of general oscillator systems to study specific models of neuronal oscillators it is shown that the degree of stochastic synchronization is controlled both by the firing rate of the neuron and the membership of the prc type i or type ii it is also shown that the circular variance for the integrate and fire neuron and the generalized order parameter for a hippocampal interneuron model have a nonlinear relationship to the input correlation

the p variation of partial sum processes and the empirical process the p variation of a function f is the supremum of the sums of the p th powers of absolute increments of f over nonoverlapping intervals let f be a continuous probability distribution function dudley has shown that the p variation of the empirical process is bounded in probability as n to infty if and only if p 2 and for 1 leq p leq 2 the p variation of the empirical process is at least n 1 p 2 and is at most of the order n 1 p 2 log log n p 2 in probability in this paper we prove that the exact order of the 2 variation of the empirical process is log log n in probability and for 1 leq p 2 the p variation of the empirical process is of exact order n 1 p 2 in expectation and almost surely let s j x 1 x 2 dots x j then the p variation of the partial sum process for x 1 x 2 dots x n is defined as that of f on 0 n where f t s j for j 1 t leq j j 1 2 dots n bretagnolle has shown that the expectation of the p variation for independent centered random variables x i with bounded p th moments is of order n for 1 leq p 2 we prove that for p 2 the 2 variation of the partial sum process of i i d centered nonconstant random variables with finite 2 delta moment for some delta 0 is of exact order n log log n in probability

limit theorems for sums of weakly dependent banach space valued random variables we prove an estimate for the prohorov distance in the central limit theorem for strong mixing banach space valued random variables using a recent variant of an approximation theorem of berkes and philipp 1979 we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term t tfrac 1 2 gamma for strong mixing sequences we prove a strong invariance principle with error term o t log logt 1 2

controllability for abstract semilinear control systems with homogeneous properties this paper considers the approximate controllability for a class of abstract semilinear control systems using a method called surjectivity theorems for nonlinear operator equations under restrictiv

an almost sure invariance principle for hilbert space valued martingales we obtain an almost sure approximation of a martingale with values in a real separable hilbert space h by a suitable h valued brownian motion

the central limit theorem for riesz raikov sums ii letf be a real valued function with period 1 satisfying some regularity conditions it will be proved that for any 1 d 1 the distribution of n 1 2 sum nolimits k 1 n left f theta 1 k t f theta d k t right on the probability space 0 1 dt converges to a normal distribution whose covariance is given by algebraic relations among the i s this generalizes the classical work by m kac and refines the characterization off to have a degenerate limit it also shows that the limit law of sum nolimits k 1 n f theta 1 k t sum nolimits l 1 n f theta j l t is in most cases a cauchy distribution

gaussian approximation of suprema of empirical processes we develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of gaussian processes without taking the route of approximating empirical processes themselves in the sup norm we prove an abstract approximation theorem that is applicable to a wide variety of problems primarily in statistics especially the bound in the main approximation theorem is non asymptotic and the theorem does not require uniform boundedness of the class of functions the proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors the proof of which depends on an effective use of stein s method for normal approximation and some new empirical processes techniques we study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not donsker type importantly our new technique is able to prove the gaussian approximation for the supremum type statistics under considerably weak regularity conditions especially concerning the bandwidth and the number of series functions in those examples

approximate controllability of semilinear functional equations in hilbert spaces in this paper approximate and complete controllability for semilinear functional differential systems is studied in hilbert spaces sufficient conditions are established for each of these types of controllability the results address the limitation that linear systems in infinite dimensional spaces with compact semigroup cannot be completely controllable the conditions are obtained by using the schauder fixed point theorem when the semigroup is compact and the banach fixed point theorem when the semigroup is not compact

synchronization and desynchronization of self sustained oscillators by common noise we consider the effect of external noise on the dynamics of limit cycle oscillators the lyapunov exponent becomes negative under influence of small white noise what means synchronization of two or more identical systems subject to common noise we analytically study the effect of small nonidentities in the oscillators and in the noise and derive statistical characteristics of deviations from the perfect synchrony large white noise can lead to desynchronization of oscillators provided they are nonisochronous this is demonstrated for the van der pol duffing system

metric marginal problems for set valued or non measurable variables in a separable metric space if two borel probability measures laws are nearby in a suitable metric then there exist random variables with those laws which are nearby in probability specifically by a well known theorem of strassen the prohorov distance between two laws is the infimum of ky fan distances of random variables with those laws the present paper considers possible extensions of strassen s theorem to two random elements one of which may be compact set valued and or non measurable there are positive results in finite dimensional spaces but with factors depending on the dimension examples show that such factors cannot entirely be avoided so that the extension of strassen s theorem to the present situation fails in infinite dimensions

nonparametric spatial regression under near epoch dependence abstract this paper establishes asymptotic normality and uniform consistency with convergence rates of the local linear estimator for spatial near epoch dependent ned processes the class of the ned spatial processes covers important spatial processes including nonlinear autoregressive and infinite moving average random fields which generally do not satisfy mixing conditions apart from accommodating a larger class of dependent processes the proposed asymptotic theory allows for triangular arrays of heterogeneous random fields located on unevenly spaced lattices and sampled over regions of arbitrary configuration all these features make the results applicable in a wide range of empirical settings

jackknife multiplier bootstrap finite sample approximations to the u process supremum with applications this paper is concerned with finite sample approximations to the supremum of a non degenerate u process of a general order indexed by a function class we are primarily interested in situations where the function class as well as the underlying distribution change with the sample size and the u process itself is not weakly convergent as a process such situations arise in a variety of modern statistical problems we first consider gaussian approximations namely approximate the u process supremum by the supremum of a gaussian process and derive coupling and kolmogorov distance bounds such gaussian approximations are however not often directly applicable in statistical problems since the covariance function of the approximating gaussian process is unknown this motivates us to study bootstrap type approximations to the u process supremum we propose a novel jackknife multiplier bootstrap jmb tailored to the u process and derive coupling and kolmogorov distance bounds for the proposed jmb method all these results are non asymptotic and established under fairly general conditions on function classes and underlying distributions key technical tools in the proofs are new local maximal inequalities for u processes which may be useful in other problems we also discuss applications of the general approximation results to testing for qualitative features of nonparametric functions based on generalized local u processes

type ii phase resetting curve is optimal for stochastic synchrony the phase resetting curve prc describes the response of a neural oscillator to small perturbations in membrane potential its usefulness for predicting the dynamics of weakly coupled deterministic networks has been well characterized however the inputs to real neurons may often be more accurately described as barrages of synaptic noise effective connectivity between cells may thus arise in the form of correlations between the noisy input streams we use constrained optimization and perturbation methods to prove that the prc shape determines susceptibility to synchrony among otherwise uncoupled noise driven neural oscillators prcs can be placed into two general categories type i prcs are non negative while type ii prcs have a large negative region here we show that oscillators with type ii prcs receiving common noisy input synchronize more readily than those with type i prcs

stimulus dependent correlations and population codes the magnitude of correlations between stimulus driven responses of pairs of neurons can itself be stimulus dependent we examine how this dependence affects the information carried by neural populations about the stimuli that drive them stimulus dependent changes in correlations can both carry information directly and modulate the information separately carried by the firing rates and variances we use fisher information to quantify these effects and show that although stimulus dependent correlations often carry little information directly their modulatory effects on the overall information can be large in particular if the stimulus dependence is such that correlations increase with stimulus induced firing rates this can significantly enhance the information of the population when the structure of correlations is determined solely by the stimulus however in the presence of additional strong spatial decay of correlations such stimulus dependence may have a negative impact opposite relationships hold when correlations decrease with firing rates

strong approximation results for the empirical process of stationary sequences we prove a strong approximation result for the empirical process associated to a stationary sequence of real valued random variables under dependence conditions involving only indicators of half lines this strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero as soon as the correlations decrease more rapidly than n 1 for some positive this shows that our conditions are in some sense optimal

nearby variables with nearby conditional laws and a strong approximation theorem for hilbert space valued martingales in this paper we focus on sequences of random vectors which do not admit a strong approximation of their partial sums by sums of independent random vectors in the first part we prove conditional versions of the strassen dudley theorem we apply these in the second part of the paper to obtain strong invariance principles for vector valued martingales which when properly normalized converge in law to a mixture of gaussian distributions

intersection bounds estimation and inference we develop a practical and novel method for inference on intersection bounds namely bounds defined by either the infimum or supremum of a parametric or nonparametric function or equivalently the value of a linear programming problem with a potentially infinite constraint set we show that many bounds characterizations in econometrics for instance bounds on parameters under conditional moment inequalities can be formulated as intersection bounds our approach is especially convenient for models comprised of a continuum of inequalities that are separable in parameters and also applies to models with inequalities that are nonseparable in parameters since analog estimators for intersection bounds can be severely biased in finite samples routinely underestimating the size of the identified set we also offer a median bias corrected estimator of such bounds as a by product of our inferential procedures we develop theory for large sample inference based on the strong approximation of a sequence of series or kernel based empirical processes by a sequence of penultimate gaussian processes these penultimate processes are generally not weakly convergent and thus are non donsker our theoretical results establish that we can nonetheless perform asymptotically valid inference based on these processes our construction also provides new adaptive inequality moment selection methods we provide conditions for the use of nonparametric kernel and series estimators including a novel result that establishes strong approximation for any general series estimator admitting linearization which may be of independent interest

some results on the lil in banach space with applications to weighted empirical processes we examine the cluster set of s n a n for banach space valued random variables and investigate the relationship between the central limit theorem and the law of the iterated logarithm in this setting in the case of hilbert space valued random variables necessary and sufficient conditions are given for the law of the iterated logarithm some interesting examples are also included we then apply our results to weighted empiricals both in the supremum norm and the l 2 lbrack 0 1 rbrack norm

controllability properties of linear and semilinear abstract control systems we consider control systems described by a semilinear abstract equation y ay f y u bu in hilbert space general conditions for exact reachability and approximate controllability are given which are related with two families of associated quadratic optimal control problems the minimum norm optimal control and the construction of the reachable set of the corresponding linear system y ay bu are characterized respectively exact reachability for a class of semilinear control systems is obtained under some assumptions on the range of a nonlinear operator mathcal f t 0 t see definition in 4 generated by the nonlinear function f

averaging principle for a class of stochastic reaction diffusion equations we consider the averaging principle for stochastic reaction diffusion equations under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component we calculate limiting slow motion the study of solvability of kolmogorov equations in hilbert spaces and the analysis of regularity properties of solutions allow to generalize the classical approach to finite dimensional problems of this type in the case of spde s

correlation transfer in stochastically driven neural oscillators over long and short time scales in the absence of synaptic coupling two or more neural oscillators may become synchronized by virtue of the statistical correlations in their noisy input streams recent work has shown that the degree of correlation transfer from input currents to output spikes depends not only on intrinsic oscillator dynamics but also on the length of the observation window over which the correlation is calculated in this paper we use stochastic phase reduction and regular perturbations to derive the correlation of the total phase elapsed over long time scales a quantity that provides a convenient proxy for the spike count correlation over short time scales we derive the spike count correlation directly using straightforward probabilistic reasoning applied to the density of the phase difference our approximations show that output correlation scales with the autocorrelation of the phase resetting curve over long time scales we also find a concise expression for the influence of the shape of the phase resetting curve on the initial slope of the output correlation over short time scales these analytic results together with numerical simulations provide new intuitions for the recent counterintuitive finding that type i oscillators transfer correlations more faithfully than do type ii over long time scales while the reverse holds true for the better understood case of short time scales

higher order statistics of input ensembles and the response of simple model neurons pairwise correlations among spike trains recorded in vivo have been frequently reported it has been argued that correlated activity could play an important role in the brain because it efficiently modulates the response of a postsynaptic neuron we show here that a neuron s output firing rate critically depends on the higher order statistics of the input ensemble we constructed two statistical models of populations of spiking neurons that fired with the same rates and had identical pairwise correlations but differed with regard to the higher order interactions within the population the first ensemble was characterized by clusters of spikes synchronized over the whole population in the second ensemble the size of spike clusters was on average proportional to the pairwise correlation for both input models we assessed the role of the size of the population the firing rate and the pairwise correlation on the output rate of two simple model neurons a continuous firing rate model and a conductance based leaky integrate and fire neuron an approximation to the mean output rate of the firing rate neuron could be derived analytically with the help of shot noise theory interestingly the essential features of the mean response of the two neuron models were similar for both neuron models the three input parameters played radically different roles with respect to the postsynaptic firing rate depending on the interaction structure of the input for instance in the case of an ensemble with small and distributed spike clusters the output firing rate was efficiently controlled by the size of the input population in addition to the interaction structure the ratio of inhibition to excitation was found to strongly modulate the effect of correlation on the postsynaptic firing rate

qualitative changes in phase response curve and synchronization at the saddle node loop bifurcation prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation the bogdanov takens bt point this study unveils another relevant and so far underestimated transition point the saddle node loop bifurcation which can be reached by several parameters including capacitance leak conductance and temperature this bifurcation turns out to induce even more drastic changes in synchronization than the bt transition this result arises from a direct effect of the saddle node loop bifurcation on the limit cycle and hence spike dynamics in contrast the bt bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking we specifically demonstrate that the saddle node loop bifurcation i ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation and ii results in a symmetry breaking of the system s phase response curve the latter entails an increase in synchronization range in pulse coupled oscillators such as neurons the derived bifurcation structure is of interest in any system for which a relaxation limit is admissible such as josephson junctions and chemical oscillators

approximate controllability for a class of semilinear abstract equations in this paper a class of control systems governed by the semilinear abstract equation dot y ay f y bv is considered a sufficient condition for the approximate controllability is obtained which improves j henry s results on a nonlinear parabolic control systems with some more serious restrictions it is suitable not only to the infinite dimensional case but also to the finite dimensional case two examples are given to explain the applications of the theory

strong convergence in stochastic averaging principle for two time scales stochastic partial differential equations abstract the theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time scales and is relatively mature for stochastic ordinary differential equations in this paper we study the averaging principle for a class of stochastic partial differential equations with two separated time scales driven by scalar noises under suitable assumptions it is shown that the slow component strongly converges to the solution of the corresponding averaged equation

on empirical likelihood statistical functions abstract we consider the empirical likelihood method for estimation of distribution and quantile functions where side information is incorporated through moment conditions we systematically study the asymptotic properties of the estimators such as the uniform strong laws of large numbers and weak convergence over classes of functions two monte carlo examples are also given to illustrate the practical utility of the method

slow manifold and averaging for slow fast stochastic differential system abstract we consider stochastic dynamical systems with multiple time scales an intermediate reduced model is obtained and explored for a slow fast system when the fast mode is driven by white noise first and for later comparison one approximation to the reduced stochastic system on the exponentially attracting stochastic slow manifold is derived to errors of order o second because the noise only drives the fast modes averaging derives an autonomous deterministic system with errors of order o then a martingale argument accounts for fluctuations about the averaged system to form an intermediate reduced model with errors of order o the autonomous deterministic system now driven by white noise this intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold these results not only connect averaging with the stochastic slow manifold they also provide a martingale method for improving averaged models of stochastic systems

some new tests for multivariate normality a family of statistics is presented that can be used for testing goodness of fit to a parametric family these statistics include mardia s measure of multivariate kurtosis and moore and stubblebine s test for multivariate normality the asymptotic distribution of the statistics is found under mild hypotheses on the parametric family and in the case of multivariate normality the distribution is shown to be independent of the true parameter a class of tests for multivariate normality is presented and the performance of two such tests in the bivariate case is found in simulations

dependence of neuronal correlations on filter characteristics and marginal spike train statistics correlated neural activity has been observed at various signal levels e g spike count membrane potential local field potential eeg fmri bold most of these signals can be considered as superpositions of spike trains filtered by components of the neural system synapses membranes and the measurement process it is largely unknown how the spike train correlation structure is altered by this filtering and what the consequences for the dynamics of the system and for the interpretation of measured correlations are in this study we focus on linearly filtered spike trains and particularly consider correlations caused by overlapping presynaptic neuron populations we demonstrate that correlation functions and statistical second order measures like the variance the covariance and the correlation coefficient generally exhibit a complex dependence on the filter properties and the statistics of the presynaptic spike trains we point out that both contributions can play a significant role in modulating the interaction strength between neurons or neuron populations in many applications the coherence allows a filter independent quantification of correlated activity in different network models we discuss the estimation of network connectivity from the high frequency coherence of simultaneous intracellular recordings of pairs of neurons

a note on nearby variables with nearby conditional laws we apply a version of the strassen dudley theorem for markov kernels to obtain a strong approximation theorem for sequences of random variables with values in polish spaces the conditions are expressed in terms of the prohorov distance of regular conditional distributions by avoiding discrete approximation of the random variables we can improve the upper bounds for the approximation

the central limit theorem and the law of iterated logarithm for empirical processes under local conditions a clt and a lil are proved under weak l2 gaussian bracketing conditions weaker than the usual ones these results have wide applicability and in particular provide an improvement of the jain marcus central limit theorem for c s valued random variables

synchrony of limit cycle oscillators induced by random external impulses the mechanism of phase synchronization between uncoupled limit cycle oscillators induced by common random impulsive forcing is analyzed by reducing the dynamics of the oscillator to a random phase map it is shown that phase synchronization generally occurs when the oscillator is driven by weak random impulsive forcing in the limit of large interimpulse intervals the case where the interimpulse intervals are finite is also analyzed perturbatively for small impulse intensity for weak poisson impulses it is shown that the phase synchronization persists up to the first order approximation

approximate controllability of semilinear neutral evolution systems with delay this paper considers the approximate controllability for a class of neutral control systems governed by semilinear neutral delayed equations sufficient conditions for approximate controllability are established by constructing fundamental solutions and using resolvent condition banach contraction mapping principle and techniques on fractional power operators the obtained results improve some existing analogous ones on this topic to some extent an example is presented finally to illustrate the applications of the obtained results

strong averaging principle for two time scale non autonomous stochastic fitzhugh nagumo system with jumps in this paper we study an averaging principle for stochastic fitzhugh nagumo system with different time scales driven by cylindrical wiener processes and poisson jumps where the slow equation is non autonomous and the fast equation is autonomous case under suitable assumptions we show that the slow component mean square strongly converges to the solution of the corresponding averaging equation and the rate of the convergence as a by product is also affirmed finally we give some open problems which are derived from this paper

strong approximations for partial sums of i i d b valued r v s in the domain of attraction of a gaussian law we obtain a strong approximation theorem for partial sums of i i d d dimensional r v s with possibly infinite second moments using this result we can extend philipp s strong invariance principle for partial sums of i i d b valued r v s satisfying the central limit theorem tob valued r v s which are only in the domain of attraction of a gaussian law this new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of kuelbs 1985

average and deviation for slow fast stochastic partial differential equations abstract averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes this article derives an averaged equation for a class of stochastic partial differential equations without any lipschitz assumption on the slow modes the rate of convergence in probability is obtained as a byproduct importantly the stochastic deviation between the original equation and the averaged equation is also studied a martingale approach proves that the deviation is described by a gaussian process this gives an approximation to errors of order o instead of order o attained in previous averaging

controllability of semilinear control systems dominated by the linear part while various equivalent conditions for controllability have been obtained in the case of linear control systems controllability problems of semilinear control systems usually require some complicated and limited assumptions in this paper we show the approximate controllability of an abstract semilinear control system under the assumption which has a simple form and can be easily checked in many examples

strong diffusion approximations for recursive stochastic algorithms lai and robbins 1978 prove strong diffusion approximations for the robbins monro stochastic approximation algorithm we show that similar strong approximations hold for stochastic algorithms at the level of generality proposed in the monograph of benveniste metivier and priouret 1990 wherein algorithms with generally discontinous right hand sides driven by conditionally markovian data are considered the relevance of our result is demonstrated on an estimation algorithm with a discontinuous right hand side which is used in data communication the technique of adaptive delta modulation used in digital communication is considered

invariance principles for parabolic equations with random coefficients a general hilbert space based stochastic averaging theory is brought forth herein for arbitrary order parabolic equations with possibly long range dependent random coefficients we use regularity conditions on tue t x 0 k 2pak t e x kxue t x ue 0 x x 1 which are slightly stronger than those required to prove pathwise existence and uniqueness for 1 equation 1 can be obtained from the singularly perturbed system ve x 0 k 2peak x kxve x ve 0 x x 2 through time change next we impose on the coefficients of 1 a pointwise inxandt weak law of large numbers and a weak invariance principleeh te 10ak x s a0k x ds k 2p k k 2p 3 inc 0 t h1 h1being a separable hilbert space of functions andh 0 1 denoting a constant h 1 2 allows for long range time dependence then under these extraordinarily general conditions we infer the weak invariance principleeh 1 ue u uis the non random e homogeneous solution of tu t x 0 k 2pa0k x kxu t x u 0 x x 4 and mildly satisfies the linear stochastic partial differential equation ty t x k 2pa0k x kxy t x dt k 2p k dt x kxu t x 5

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes

control problems for semi linear retarded integro differential equations by the fredholm theory in this paper we deal with the approximate controllability for semi linear retarded functional integro differential equations by using the fredholm theory in hilbert spaces we no longer require t