extremes of moving averages of random variables with finite endpoint where zj are iid and nonnegative random variables and c 0 are constants satisfying summability conditions at least sufficient to make the random series above converge we suppose that the distribution of zj is regularly varying near 0 and discuss lower tail behavior of finite and infinite linear combinations the behavior is quite different in the two cases for finite linear combinations the lower tail is again regularly varying but for infinite moving averages the lower tail is f varying i e it is in the domain of attraction of a type i extreme value distribution in the sense of minima convergence of point processes based on the moving averages is shown to hold in both the finite and infinite order cases and suitable conclusions are drawn from such convergences a useful analytic tool is asymptotic normality of the esscher transform of the common distribution of the z s the extreme value results of this paper are in terms of minima of the moving average processes but results can be adapted to study maxima of moving averages of random variables in the domain of attraction of the type iii extreme value distribution for maxima

mass concentration and aging in the parabolic anderson model with doubly exponential tails we study the non negative solution u u x t to the cauchy problem for the parabolic equation partial t u vardelta u xi u on mathbb z d times 0 infty with initial data u x 0 mathsf 1 0 x here vardelta is the discrete laplacian on mathbb z d and xi xi z z in mathbb z d is an i i d random field with doubly exponential upper tails we prove that for large t and with large probability most of the total mass u t sum x u x t of the solution resides in a bounded neighborhood of a site z t that achieves an optimal compromise between the local dirichlet eigenvalue of the anderson hamiltonian vardelta xi and the distance to the origin the processes t mapsto z t and t mapsto tfrac 1 t log u t are shown to converge in distribution under suitable scaling of space and time aging results for z t as well as for the solution to the parabolic problem are also established the proof uses the characterization of eigenvalue order statistics for vardelta xi in large sets recently proved by the first two authors

basis functions and parameter optimisation in high order iterative learning control in this paper a new parameter optimal high order iterative learning control ilc algorithms is proposed to extend the work of owens and feng parameter optimisation in iterative learning control international journal of control 14 11 1059 1069 if the original plant is positive this new algorithm will result in convergent learning where the convergence is monotonic to zero tracking error if the original plant is not positive it can be shown that by adding a suitable set of basis functions into the algorithm the tracking error will again converge monotonically to zero this provides a considerable improvement to earlier work on parameter optimal ilc as it opens up the possibility of globally convergent algorithms for any linear plant g the number of parameters needed to ensure convergence could however become large the paper shows that the use of low order parameterisations is capable of achieving much of the benefit achieved in the ideal case

long time tails in the parabolic anderson model with bounded potential we consider the parabolic anderson problem t u u u on 0 z d with random i i d potential z z zd and the initial condition u 0 1 our main assumption is that esssup 0 0 depending on the thickness of the distribution prob 0 close to its essential supremum we identify both the asymptotics of the moments of u t 0 and the almost sure asymptotics of u t 0 as t in terms of variational problems as a by product we establish lifshitz tails for the random schrodinger operator at the bottom of its spectrum in our class of distributions the lifshitz exponent ranges from d 2 to the power law is typically accompanied by lower order corrections

extreme value theory for moving average processes this paper studies extreme values in infinite moving average processes x t sum lambda c lambda t z lambda defined from an i i d noise sequence z lambda in particular this includes the arma processes often used in time series analysis a fairly complete extremal theory is developed for the cases when the d f of the z lambda s has a smooth tail which decreases approximately as exp z p as z rightarrow infty for 0 p infty or as a power of z the influence of the averaging on extreme values depends on p and the c lambda s in a rather intricate way for p 2 which includes normal sequences the correlation function r t sum lambda c lambda t c lambda sum lambda c 2 lambda determines extremal behavior while perhaps more surprisingly for p neq 2 correlations have little bearing on extremes further the sample paths of x t near extreme values asymptotically assume a specific nonrandom form which again depends on p and c lambda in an interesting way one use of this latter result is as an informal quantitative check of a fitted moving average or arma model by comparing the sample path behavior predicted by the model with the observed sample paths

a survey on iterative learning control for nonlinear systems in this article we review the recent advances in iterative learning control ilc for nonlinear dynamic systems in the research field of ilc two categories of system nonlinearities are considered namely the global lipschitz continuous glc functions and local lipschitz continuous llc functions ilc for glc systems is widely studied and analysed using contraction mapping approach and the focus of recent exploration moves to application problems though a number of theoretical issues remain open ilc for llc systems is currently a hot area and the recent research focuses on ilc design and analysis by means of lyapunov approach the objectives of this article are to introduce recent development and advances in nonlinear ilc schemes highlight their effectiveness and limitations as well as discuss the directions for further exploration of nonlinear ilc

critical exponents of directed self avoiding walks it is argued that directed self avoiding random walks are in the same universality class as directed unrestricted random walks and therefore should exhibit mean field exponents gamma 1 nu sub 1 nu perpendicular to 1 2 in all dimensionalities

the universality classes in the parabolic anderson model we discuss the long time behaviour of the parabolic anderson model the cauchy problem for the heat equation with random potential on mathbb z d we consider general i i d potentials and show that exactly four qualitatively different types of intermittent behaviour can occur these four universality classes depend on the upper tail of the potential distribution 1 tails at that are thicker than the double exponential tails 2 double exponential tails at studied by gartner and molchanov 3 a new class called almost bounded potentials and 4 potentials bounded from above studied by biskup and konig the new class 3 which contains both unbounded and bounded potentials is studied in both the annealed and the quenched setting we show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time the characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas respectively gaussian densities our analysis of class 3 relies on two large deviation results for the local times of continuous time simple random walk one of these results is proved by brydges and the first two authors in bhk04 and is also used here to correct a proof in bk01

multivariable norm optimal and parameter optimal iterative learning control a unified formulation this article investigates the two paradigms of norm optimal iterative learning control noilc and parameter optimal iterative learning control poilc for multivariable mimo l input m output linear discrete time systems the main result is a proof that despite their algebraic and conceptual differences they can be unified using linear quadratic multi parameter optimisation techniques in particular whilst poilc has been naturally regarded as an approximation to noilc it is shown that the noilc control law can be generated from a suitable choice of control law parameterisation and objective function in a multi parameter mimo poilc problem the form of this equivalence is used to propose a new general approach to the construction of poilc problems for mimo systems that approximates the solution of a given noilc problem an infinite number of such approximations exist this great diversity is illustrated by the derivation of new convergent algorithms based on time interval and gradient partition that

critical properties of directed self avoiding walks the generating functions and mean displacements of various two dimensional directed self avoiding walk models are calculated exactly by a simple transfer matrix approach asymptotically the authors find r sub n approximately rn2 1 2 approximately n and rperpendicular to n2 1 approximately n1 2 where n is the number of steps in the walk and and perpendicular to refer to projections of the displacement parallel and perpendicular to the preferred axis of the walk respectively some general properties of directed self avoiding walks for arbitrary dimensions are discussed as well

renormalisation group study of fully directed self avoiding walks the author presents a discussion of the critical properties of fully directed self avoiding walks on hypercubic lattices a position space renormalisation group transformation is carried out analytically in two dimensions for any value of the scaling factor b the finite size scaling properties of the transformation are explicitly exhibited including the well known finite size scaling hypothesis v b v approximately 1n b 1 it is shown that the rg transformation in this case is sensitive to v sub thus giving support to existing arguments

performance of cusum control schemes for serially correlated observations this article discusses situations in which one is interested in evaluating the run length characteristics of a cumulative sum control scheme when the underlying data show the presence of serial correlation in practical applications situations of this type are common in problems associated with monitoring such characteristics of data as forecasting errors measures of model adequacy and variance components the discussed problem is also relevant in situations in which data transformations are used to reduce the magnitude of serial correlation the basic idea of analysis involves replacing the sequence of serially correlated observations by a sequence of independent and identically distributed observations for which the runlength characteristics of interest are roughly the same applications of the proposed method for several classes of processes arising in the area of statistical process control are discussed in detail and it is shown that it leads to approximations that can be considered acceptable in

on the extremes of a class of non linear processes with heavy tailed innovations the aim of this paper is to look at the limiting form of certain empirical point processes induced by a particular class of non linear processes generated by heavy tailed innovations such asymptotic results will be highly useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics in particular we investigate the maximum limiting distribution and its corresponding extremal index the results are applied to the study of the extremal properties of bilinear processes

a short note on directed self avoiding walks the critical behaviour of partially directed self avoiding walks saws on a cubic lattice with only one axis directed is studied using exact enumerations the critical exponents nu sub 0 999 or 0 005 nu perpendicular to 0 501 or 0 006 and gamma 1 000 or 0 001 found here further support the universality hypothesis that various partially directed saws belong to the same universality class as fully directed saws and they are also dimension independent for d or 2

joint density for the local times of continuous time markov chains we investigate the local times of a continuous time markov chain on an arbitrary discrete state space for fixed finite range of the markov chain we derive an explicit formula for the joint density of all local times on the range at any fixed time we use standard tools from the theory of stochastic processes and finite dimensional complex calculus we apply this formula in the following directions 1 we derive large deviation upper estimates for the normalized local times beyond the exponential scale 2 we derive the upper bound in varadhan s lemma for any measurable functional of the local times and 3 we derive large deviation upper bounds for continuous time simple random walk on large subboxes of z d tending to z d as time diverges we finally discuss the relation of our density formula to the ray knight theorem for continuous time simple random walk on z which is analogous to the well known ray knight description of brownian local times

application of directed lattice animal theory to river networks it is shown that the theory of directed lattice animals is a good candidate for determining the exponents which describe river networks

localized growth and branching random walks with time correlations we generalize a model of growth over a disordered environment to a large class of it processes in particular we study how the microscopic properties of the noise influence the macroscopic growth rate the present model can account for growth processes in large dimensions and provides a bed to understand better the tradeoff between exploration and exploitation an additional mapping to the schrodinger equation readily provides a set of disorders for which this model can be solved exactly this mean field approach exhibits interesting features such as a freezing transition and an optimal point of growth which can be studied in detail and gives yet another explanation for the occurrence of the zipf law in complex well connected systems

large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes in this paper we prove exact forms of large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes we also show that a fractional brownian motion and the related riemann liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times as a consequence of our large deviation estimates we derive laws of iterated logarithm for the corresponding local times the key points of our methods 1 logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional brownian motion 2 logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of riemann liouville processes 3 comparison of local and intersection local times based on embedding of a part of a fractional brownian motion into the reproducing kernel hilbert space of the riemann liouville process

strong laws for the k th order statistic when k c log2n we give several bounds and characterizations for the lower almost sure classes of uk n where k kn denotes a nondecreasing integer sequence such that 1 kn 0 log2n as n and where u1 n un n denote the order statistics of the first n observations from a sequence of independent and uniformly distributed random variables on 0 1

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

geometric characterization of intermittency in the parabolic anderson model we consider the parabolic anderson problem tu u x u on r z d with localized initial condition u 0 x 0 x and random i i d potential under the assumption that the distribution of 0 has a double exponential or slightly heavier tail we prove the following geometric characterisation of intermittency with probability one as t 1 the overwhelming contribution to the total mass p x u t x comes from a slowly increasing number of islands which are located far from each other these islands are local regions of those high exceedances of the eld in a box of side length 2t log 2 t for which the local principal dirichlet eigenvalue of the random operator is close to the top of the spectrum in the box we also prove that the shape of in these regions is non random and that u t is close to the corresponding positive eigenfunction this is the geometric picture suggested by localization theory for the anderson hamiltonian

correlation structure of intermittency in the parabolic anderson model consider the cauchy problem u x t t u x t x d t 0 with initial condition u x 0 1 and with the anderson hamiltonian here is the discrete laplacian 0 is a diffusion constant and x x d is an i i d random field taking values in gartner and molchanov 1990 have shown that if the law of 0 is nondegenerate then the solution u is asymptotically intermittent in the present paper we study the structure of the intermittent peaks for the special case where the law of 0 is in the vicinity of the double exponential prob 0 s exp e s s here 0 is a parameter that can be thought of as measuring the degree of disorder in the field our main result is that for fixed x y d and t the correlation coefficient of u x t and u y t converges to w 2 l2 z d w x z w y z in this expression while w d is given by w v d with v the unique centered ground state i e the solution in l2 with minimal l 2 norm of the 1 dimensional nonlinear equation v 2 v log v 0 the uniqueness of the ground state is actually proved only for large but is conjectured to hold for any 0 empty it turns out that if the right tail of the law of 0 is thicker or thinner than the double exponential then the correlation coefficient of u x t and u y t converges to x y resp the constant function 1 thus the double exponential family is the critical class exhibiting a nondegenerate correlation structure

ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential in this paper we show the that the ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential is controlled asymptotically as the system size n goes to infinity by the random variable l n the length the longest consecutive sequence of sites on the lattice with potential equal to zero specifically we will show that for almost every realization of the potential the ground state energy behaves asymptotically as frac pi 2 ell n 1 2 in the sense that the ratio of the quantities goes to one

parameter optimal iterative learning control using polynomial representations of the inverse plant based on the observation that iterative learning control ilc can be based on the inverse plant but that the approach can be degraded by modelling errors particularly at high frequencies this article investigates the construction and properties of a multi parameter parameter optimal ilc algorithm that uses an approximate polynomial representation of the inverse with natural high frequency attenuation in its simplest form the algorithm replicates the original work of owens and feng but more generally it is capable of producing significant improvements to the observed convergence rate as the number of parameters increases convergence rates approach that of the ideal plant inverse algorithm introducing compensation into the algorithm provides a formal link to previously published gradient and norm optimal ilc algorithms and indicates that the polynomial approach can be regarded as approximations to those control laws simulation examples verify the theoretical performance predictions

parabolic problems for the anderson model the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

a limit theorem for the maximum of autoregressive processes with uniform marginal distributions a class of first order autoregressive processes is given for which the extreme value limit theorems of loynes and leadbetter do not apply a limit theorem is derived for these processes that depends on the parameter r an integer greater than or equal to 2

oracally efficient estimation of autoregressive error distribution with simultaneous confidence band we propose kernel estimator for the distribution function of unobserved errors in autoregressive time series based on residuals computed by estimating the autoregressive coefficients with the yule walker method under mild assumptions we establish oracle efficiency of the proposed estimator that is it is asymptotically as efficient as the kernel estimator of the distribution function based on the unobserved error sequence itself applying the result of wang cheng and yang j nonparametr stat 25 2013 395 407 the proposed estimator is also asymptotically indistinguishable from the empirical distribution function based on the unobserved errors a smooth simultaneous confidence band scb is then constructed based on the proposed smooth distribution estimator and kolmogorov distribution simulation examples support the asymptotic theory

critical exponent of a directed self avoiding walk the authors have investigated numerically and analytically the directed self avoiding walk problem recently proposed and studied by chakrabarti and manna 1983 they find a different result for the exponent nu namely nu 1

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

flory theory for directed lattice animals and directed percolation the free energies of directed lattice animals in good and theta solvents and the free energy of directed percolation are found by the use of a simple flory type approximation which accounts for the inherent anisotropy of these systems from these free energies the authors obtain the upper critical dimension below which mean field theory breaks down they also calculate closed form dimension dependent expressions for the parallel and perpendicular correlation length exponents which characterise the asymptotic cluster shapes these exponents are in excellent agreement with existing numerical data

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

on some estimates based on sample behavior near high level excursions let j be a stationary sequence of weakly dependent random variables and letm n k be thek th largest value of j 1 j n the estimation of the parameters of the asymptotic distribution ofm n k is considered using a procedure motivated by a limit theorem pertaining to the point process sum nolimits j delta j n n bar f xi j a number of statistical issues concerning the procedure including how to select the tuning parameters are addressed the second problem that we consider is the estimation of the filter of a moving average process with heavy tails in particular the investigation covers the moving average stable process motivated by ideas in rootzen 1978 our estimator uses information contained in the sample behavior of the process near the largest excursion

asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions we establish some asymptotic expansions for infinite weighted convolutions of distributions having rapidly varying subexponential tails examples are presented some showing that in order to obtain an expansion with two significant terms one needs to have a general way to calculate higher order expansions due to possible cancellations of terms an algebraic methodology is employed to obtain the results

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

exponent vperp for 2d directed compact site animals the authors obtain the transverse cluster radius exponent vperp 0 for 2d directed compact animals they find that the exponent vperp takes two different values which depend on definition of the animal width

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

modelling extremes of the areal rainfall process risk assessment for many hydrological structures requires an estimate of the extremal behaviour of the rainfall regime within a specified catchment region in most cases it is the spatially aggregated rainfall which is the key process though in practice only pointwise rainfall measurements from a network of sites over the region are available in this paper we address the problem of making inferences about the extremal properties of the aggregated process from the pointwise data working within the usual extreme value paradigm a model is derived in which the resulting distribution is determined by the marginal tail behaviour and spatial dependence at extreme levels of the process data collected from a region in the south west of england are used to illustrate the procedure

generalised self avoiding walk a generalisation of the self avoiding walk is introduced in which k or higher multiple points are forbidden k 2 corresponds to the standard self avoiding walk the flory theory gives the radius of gyration exponent nu k k 1 k 1 e 2 when e or ec k 2k k 1 e is the euclidean dimension of the problem and ec k the upper critical dimension which is also obtained using the fractal set theory

quenched asymptotics for brownian motion of renormalized poisson potential and for the related parabolic anderson models let bs be a d dimensional brownian motion and dx be an independent poisson field on rd the almost sure asymptotics for the logarithmic moment generating function loge0exp t0v bs ds t are investigated in connection with the renormalized poisson potential of the form v x rd1 y x p dy dy x rd the investigation is motivated by some practical problems arising from the models of brownian motion in random media and from the parabolic anderson models

phase transition for the speed of the biased random walk on the supercritical percolation cluster we prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on d that is for each d 2 and for any supercritical parameter p pc we prove the existence of a critical strength for the bias such that below this value the speed is positive and above the value it is zero we identify the value of the critical bias explicitly and in the subballistic regime we find the polynomial order of the distance moved by the particle each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk a key element in proving our results is to understand that on large scales the particle trajectory is essentially one dimensional we prove such a dynamic renormalization statement in a much stronger form than was previously known 2013 wiley periodicals inc

limit sets and switching strategies in parameter optimal iterative learning control this paper characterizes the existence and form of the possible limit error signals in typical parameter optimal iterative learning control the set of limit errors has attracting and repelling components and the behaviour of the algorithm in the vicinity of these sets can be associated with the undesirable properties of apparent but in fact temporary convergence or permanent slow convergence properties in practice the avoidance of these behaviours in practice is investigated using novel switching strategies deterministic strategies are analysed to prove the feasibility of the concept by proving that each of a number of such strategies is guaranteed to produce global convergence of errors to zero independent of the details of plant dynamics for practical applications a random switching strategy is proposed to replace these approaches and shown by example to produce substantial potential improvements when compared with the non switching case the work described in this paper is covered by pending patent applications in the uk and elsewhere

extremal theory for spectrum of random discrete schrodinger operator iii localization properties in this paper we study the asymptotic localization properties with high probability of the kth eigenfunction associated with the kth largest eigenvalue k 1 fixed of the multidimensional anderson hamiltonian in torus v increasing to the whole of lattice denote by z k v v the site at which the kth largest value of potential is attained it is well known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity then the kth eigenfunction is asymptotically delta function at the site z k v localization centre for some random k v k 1 we study the asymptotic behavior of the index v k by distinguishing between three cases of the tails of potential distribution i for the heavy tails including gaussian v k is asymptotically bounded ii for the light tails but heavier than the double exponential the index v k unboundedly increases like v o 1 iii finally for the double exponential tails with high disorder the index v k behaves like a power of v for weibull s and fractional double exponential types distributions associated with the case ii we obtain the first order expansion formulas for log v k

exponential asymptotics and law of the iterated logarithm for intersection local times of random walks let a 0 1 p denote the intersection local time of p independent d dimensional brownian motions running up to the time 1 under the conditions p d 2 2 we prove lim t 1 log p 0 1 p t d p 1 2 d p with the right hand side being identified in terms of the the best constant of the gagliardo nirenberg inequality within the scale of moderate deviations we also establish the precise tail asymptotics for the intersection local time in k 1 k p 1 n p s 1 k 1 s p k p run by the independent symmetric z d valued random walks s 1 n s p n our results apply to the law of the iterated logarithm our approach is based on feynman kac type large deviation time exponentiation moment computation and some technologies along the lines of probability in banach space as an interesting coproduct we obtain the inequality formula math in the case of random walks

self intersection local times of random walks exponential moments in subcritical dimensions fix p 1 not necessarily integer with p d 2 0 that are bounded from above possibly tending to zero the speed is identified in terms of mixed powers of t and t and the precise rate is characterized in terms of a variational formula which is in close connection to the gagliardo nirenberg inequality as a corollary we obtain a large deviation principle for lt p trt for deviation functions rt satisfying t r t gg mathbb e ell t p informally it turns out that the random walk homogeneously squeezes in a t dependent box with diameter of order t1 d to produce the required amount of self intersections our main tool is an upper bound for the joint density of the local times of the walk

external theory for stochastic processes abstract the purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes the theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts emphasis is placed on stationary cases but other important classes e g mark of sequences are included significant ideas and methods are described rather than details and in particular the nature and role of important underlying point processes such as exceedances and upcrossings are emphasized applications are given to particular classes of process e g normal moving average and connections with related theory such as convergence of sums are indicated

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

extremal theory for spectrum of random discrete schrodinger operator i asymptotic expansion formulas we consider the spectral problem for the random schrodinger operator on the multidimensional lattice torus increasing to the whole of lattice with an i i d potential anderson hamiltonian we obtain the explicit almost sure asymptotic expansion formulas for the extreme eigenvalues and eigenfunctions in the intermediate rank case provided the upper distributional tails of potential decay at infinity slower than the double exponential function for the fractional exponential tails including weibull s and gaussian distributions extremal type limit theorems for eigenvalues are proved and the strong influence of parameters of the model on a specification of normalizing constants is described in the proof we use the finite rank perturbation arguments based on the cluster expansion for resolvents

a two cities theorem for the parabolic anderson model the parabolic anderson problem is the cauchy problem for the heat equation partial derivative t u t z delta u t z xi z u t z on 0 infinity x z d with random potential xi z z is an element of z d we consider independent and identically distributed potentials such that the distribution function of z converges polynomially at infinity if u is initially localized in the origin that is if u 0 z 1 0 z we show that as time goes to infinity the solution is completely localized in two points almost surely and in one point with high probability we also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem

the asymptotic behaviour of directed self avoiding walks chakrabarti and manna 1983 have made a numerical study of directed self avoiding walks on a square lattice in which the random walker is forbidden to move upward extrapolating from an enumeration of n step walks up to n 14 they claimed that the mean end to end displacement rn behaves asymptotically for large n as nnu with nu 0 86 or 0 02 it is shown that a simple application of the generating function method yields the behaviour of n step walks of this type exactly for all n and the result nu 1 is rigorously proved

extremal theory for spectrum of random discrete schrodinger operator ii distributions with heavy tails we study the asymptotic structure of the first k largest eigenvalues k v and the corresponding eigenfunctions k v of a finite volume anderson model discrete schrodinger operator mathcal h v kappa delta v xi cdot on the multidimensional lattice torus v increasing to the whole of lattice provided the distribution function f of i i d potential satisfies condition log 1 f t o t3 and some additional regularity conditions as t for z v denote by 0 z the principal eigenvalue of the single peak hamiltonian v z z in l2 v and let lambda 0 k v be the kth largest value of the sample 0 in v we first show that the eigenvalues k v are asymptotically close to lambda 0 k v we then prove extremal type limit theorems i e poisson statistics for the normalized eigenvalues k v bv av where the normalizing constants av 0 and bv are chosen the same as in the corresponding limit theorems for lambda 0 k v the eigenfunction k v is shown to be asymptotically completely localized as v at the sites zk v v defined by lambda 0 z k v lambda 0 k v proofs are based on the finite rank in particular rank one perturbation arguments for discrete schrodinger operator when potential peaks are sparse

parameter optimization in iterative learning control in this paper parameter optimization through a quadratic performance index is introduced as a method to establish a new iterative learning control law with this new algorithm monotonic convergence of the error to zero is guaranteed if the original system is a discrete time lti system and it satisfies a positivity condition if the original system is not positive two methods are derived to make the system positive the effect of the choice of weighting parameters in the performance index on convergence rate is analysed as a result adaptive weights are introduced as a method to improve the convergence properties of the algorithm a high order version of the algorithm is also derived and its convergence analysed the theoretical findings in this paper are highlighted with simulations

brownian intersection local times upper tail asymptotics and thick points we equip the intersection of p independent brownian paths in r d d 2 with the natural measure l defined by projecting the intersection local time measure via one of the brownian motions onto the set of intersection points given a bounded domain u c r d we show that as a the probability of the event l u a decays with an exponential rate of a 1 p where is described in terms of a variational problem in the important special case when u is the unit ball in r 3 and p 2 we characterize in terms of an ordinary differential equation we apply our results to the problem of finding the hausdorff dimension spectrum for the thick points of the intersection of two independent brownian paths in r 3

darling erdős theorems for normalized sums of i i d variables close to a stable law let 1 be i i d real valued random variables and s n 1 n in the case when the distribution of is close to a stable law for some 0 1 1 2 we investigate the asymptotic behavior in distribution of the maximum of normalized sums max k 1 k 1 s k this completes the darling erdos limit theorem for the case 2

almost sure asymptotics for the continuous parabolic anderson model we consider the parabolic anderson problem t u u x u on d with initial condition u 0 x 1 here 0 is a diffusion constant and is a random homogeneous potential we concentrate on the two important cases of a gaussian potential and a shot noise poisson potential under some mild regularity assumptions we derive the second order term of the almost sure asymptotics of u t 0 as t

on the apparent failure of a flory approximation for directed linear polymers the author discusses the apparent failure of a flory approximation for directed linear polymers commenting on the details implicit in this type of picture the relationship of a standard flory viewpoint to a simplified application of fractal theory is explored and a correspondence is drawn between aspects of both theories