weak convergence of linear and quadratic forms and related statements on lp approximability abstract in this paper we obtain central limit theorems for quadratic forms of non causal short memory linear processes with independent identically distributed innovations nabeya and tanaka 1988 suggested the format which links the asymptotic distribution to integral operators in their approach integral operators had to have continuous symmetric kernels mynbaev 2001 employed the theory of approximations to get rid of the continuity requirement here we go one step further by lifting the kernel symmetry condition also we establish l p approximability of the special sequences which arise in the theory of regressions with slowly varying regressors

semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

the asymptotic behavior of quadratic forms in φ mixing random variables let z i i 1 be a linear process defined by z email protected j 0 d email protected t j with d j j 0 being a regular varying sequence of real numbers and x t

central limit theorem for linear processes iences mixing sequences or associated sequences the results are important in analyzing the asymptotical properties of some estimators as well as of linear processes 4 1 introduction let be a centered sequence of random variables k 4 and let a 1 i n be a triangular array of numbers many statistical ni procedures produce estimators of the type n 1 1 s a nn i i i1 to give an example let us consider the nonlinear regression model yx gx x where gx is an unknown function and x is the noise now we fix the design points x and we get ni

the neumann problem for higher order elliptic equations with symmetric coefficients in this paper we establish well posedness of the neumann problem with boundary data in l 2 or the sobolev space dot w 2 1 in the half space for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint this generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic rellich identity and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a lebesgue or sobolev space

a nonlinear method for robust spectral analysis a nonlinear spectral analyzer called the l p norm periodogram is obtained by replacing the least squares criterion with an l p norm criterion in the regression formulation of the ordinary periodogram in this paper we study the statistical properties of the l p norm periodogram for time series with continuous and mixed spectra we derive the asymptotic distribution of the l p norm periodogram and discover an important relationship with the so called fractional autocorrelation spectrum that can be viewed as an alternative to the power spectrum in representing the serial dependence of a random process in the frequency domain in comparison with the ordinary periodogram p 2 we show that by varying the value of p in the interval 1 2 the l p norm periodogram can strike a balance between robustness against heavy tailed noise efficiency under regular conditions and spectral leakage for time series with mixed spectra we also show that the l p norm periodogram can detect serial dependence of uncorrelated non gaussian time series that cannot be detected by the ordinary periodogram

the l p theory of divergence form elliptic operators under the dirichlet condition abstract let a be the 2 m th order elliptic operator of divergence form with bounded measurable coefficients defined in a domain of r n with smooth boundary for 1 p we regard a as a bounded operator from the l p sobolev space h 0 m p to h m p trying to extend the result which has been obtained only in the case r n or in the case m 1 to the general case we succeed in constructing the resolvent a 1 and estimating its operator norm for some when the leading coefficients are uniformly continuous applying the result for l p resolvents we show that the operator associated with a in l p generates an analytic semigroup and obtain exponential decay estimates for the heat kernel the resolvent kernel and their derivatives of order up to m 1 we also give a perturbation theorem for heat kernels

central limit theorem for stationary linear processes we establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in mcleish ann probab 5 1977 616 621 and motivated by gordin soviet math dokl 10 1969 1174 1176 in doing so we shall preserve the generality of the coefficients including the long range dependence case and we shall express the variance of partial sums in a form easy to apply ergodicity is not required

equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form we study the heat kernel of higher order elliptic operators or systems under divergence form on rn ellipticity is in the sense of garding inequality we show that for homogeneous operators gaussian upper bounds and holder regularity of the heat kernel is equivalent to local regularity of weak solutions we also show stability of such bounds under l perturbations of the coefficients or under perturbations with bounded coefficients lower order terms such a criterion allows us to obtain heat kernel bounds for operators or systems with uniformly continuous or vmo coefficients

tent space well posedness for parabolic cauchy problems with rough coefficients abstract we study the well posedness of cauchy problems on the upper half space r n 1 associated to higher order systems t u 1 m 1 div m a m u with bounded measurable and uniformly elliptic coefficients we address initial data lying in l p 1 p and bmo p spaces and work with weak solutions our main result is the identification of a new well posedness class given for p 1 by distributions satisfying m u t m p 2 where t m p 2 is a parabolic version of the tent space of coifman meyer stein in the range p 2 this holds without any further constraints on the operator and for p it provides a carleson measure characterization of bmo with non autonomous operators we also prove higher order l p well posedness previously only known for the case m 1 the uniform l p boundedness of propagators of energy solutions plays an important role in the well posedness theory and we discover that such bounds hold for p close to 2 this is a consequence of local weak solutions being locally holder continuous with values in spatial l l o c p for some p 2 what is also new for the case m 1

on second order periodic elliptic operators in divergence form we consider second order strongly elliptic operators with complex coecients in divergence form on r d we assume that the coecients are all periodic with a common period if the coecients are continu ous we derive gaussian bounds with the correct small and large time asymptotic behaviour on the heat kernel and all its holder deriva tives moreover we show that the first order riesz transforms are bounded on the lp spaces with p 2 h1 1i secondly if the coe cients are holder continuous we prove that the first order derivatives of the kernel satisfy good gaussian bounds then we establish that the second order derivatives exist and satisfy good bounds if and only if the coecients are divergence free or if and only if the second order riesz transforms are bounded finally if the third order derivatives exist with good bounds then the coecients must be constant

the lp resolvents of elliptic operators with uniformly continuous coefficients abstract we consider the strongly elliptic operator a of order 2 m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous for 1 p a is a bounded linear operator from the l p sobolev space h m p into h m p we will prove that a 1 exists in h m p for some and estimate its operator norm

the intersymbol interference channel lower bounds on capacity and channel precoding loss the discrete time additive gaussian intersymbol interference isi channel with i i d not necessarily gaussian input signals is considered several new and old lower bounds on the capacity are derived in a unified manner by assuming different front end receiver filters in particular the sampled whitened matched filter swmf and the minimum mean squared error decision feedback equalizer mmse dfe filter the features of the bounds are demonstrated and compared in several examples with binary and quaternary input signals it is also shown that the effect of an ideal post cursor or tail cancellation in an information preserving context depends primarily on the front end filter while as is well known ideal post cursor cancellation at the output of the swmf decreases the information the opposite trend is seen when an mmse dfe front filter is considered this observation reflects the basic theoretical obstacles in precoding i e ideal post cursor cancellation in the presence of a pre cursor it is used to assess the inherent loss in terms of information rates as compared to the rates achievable with the hypothetical ideal post cursor cancellation associated with any post cursor cancellation technique such as precoding dfe or other variants when operating in synergy with the mmse dfe front end filter the effect of the front end filter on an ideally interleaved precoded coded modulation system is also addressed

variance estimation in nonparametric regression via the difference sequence method consider a gaussian nonparametric regression problem having both an unknown mean function and unknown variance function this article presents a class of difference based kernel estimators for the variance function optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized and asymptotic normality is also established we also show that for suitable asymptotic formulations our estimators achieve the minimax rate

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

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

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

a new maximal inequality and invariance principle for stationary sequences we derive a new maximal inequality for stationary sequences under a martingale type condition introduced by maxwell and woodroofe ann probab 28 2000 713 724 then we apply it to establish the donsker invariance principle for this class of stationary sequences a markov chain example is given in order to show the optimality of the conditions imposed

the lp resolvents of second order elliptic operators of divergence form under the dirichlet condition abstract let a be the 2mth order elliptic operator of divergence form with bounded measurable coefficients defined in a domain of r n for 1 h 0 m p to h m p it is known that when r n we can construct the resolvent a 1 and estimate its operator norm for some if the leading coefficients are uniformly continuous in this paper we try to extend this result to a general domain it is successful when m 1 if is the half space or a domain with c2 bounded boundary for m 1 it is shown that the problem is reduced to the case where is the half space and a is a homogeneous operator with constant coefficients we also give a perturbation theorem

convergence rates and interior estimates in homogenization of higher order elliptic systems abstract this paper is concerned with the quantitative homogenization of 2m order elliptic systems with bounded measurable rapidly oscillating periodic coefficients we establish the sharp o e convergence rate in w m 1 p 0 with p 0 2 d d 1 in a bounded lipschitz domain in r d as well as the uniform large scale interior c m 1 1 estimate with additional smoothness assumptions the uniform interior c m 1 1 w m p and c m 1 estimates are also obtained as applications of the regularity estimates we establish asymptotic expansions for fundamental solutions

explicit estimates on the fundamental solution of higher order parabolic equations with measurable coefficients for some positive constants ci and all t 0 and x y 0 such estimates for uniformly elliptic operators have been proved under a variety of local regularity assumptions 14 17 and more recently for operators with measurable coefficients provided 2m n 8 this has been shown to be true in the limit case 2m n 3 11 while there are examples that show that 1 is not in general true when 2m n and h has measurable coefficients 10 in 6 estimates are obtained for singular or degenerate operators with measurable coefficients with the euclidean distance being replaced by one that reflects the behavior of the coefficients near the singularities we refer to 9 for a detailed account of much of the recent progress on the spectral theory of higher order elliptic operators the problem of sharpening such bounds was addressed recently the sharpness of estimates is checked by comparison against short time asymptotics of k t x y these were proved by tintarev 19 who showed following work of evgrafov and postnikov 12 that if 0 r if the coefficients a are smooth and if the symbol a x satisfies the strong doi 10 1006 jdeq 2000 3940 available online at http www idealibrary com on

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

higher order elliptic and parabolic systems with variably partially bmo coefficients in regular and irregular domains abstract the solvability in sobolev spaces is proved for divergence form complex valued higher order parabolic systems in the whole space on a half space and on a reifenberg flat domain the leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder the directions in which the coefficients are only measurable vary depending on each cylinder the corresponding elliptic problem is also considered

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio

an almost sure central limit theorem for products of sums of partial sums under association abstract let x n n 1 be a strictly stationary positively or negatively associated sequence of positive random variables with e x 1 0 and var x 1 2 denote s n i 1 n x i t n i 1 n s i and the coefficient of variation under suitable conditions we show that x lim n 1 log n k 1 n 1 k i 2 k j 1 k t j k k 1 k 1 1 k x f x a s where 1 2 1 2 2 j 2 cov x 1 x j f is the distribution function of the random variables e 10 3 n and n is a standard normal random variable

convergence results for a class of spectrally hyperviscous models of 3 d turbulent flow abstract we consider the spectrally hyperviscous navier stokes equations shnse which add hyperviscosity to the nse but only to the higher frequencies past a cutoff wavenumber m 0 in guermond and prudhomme 2003 18 subsequence convergence of shnse galerkin solutions to dissipative solutions of the nse was achieved in a specific spectral vanishing viscosity setting our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of leray solutions in particular we obtain subsequence convergence of shnse strong solutions to leray solutions of the nse by fixing the hyperviscosity coefficient while the spectral hyperviscosity cutoff m 0 goes to infinity this formulation presents new technical challenges and we discuss how its motivation can be derived from computational experiments e g those in borue and orszag 1996 1998 3 4 we also obtain weak subsequence convergence to leray weak solutions under the general assumption that the hyperviscous coefficient goes to zero with no constraints imposed on the spectral cutoff in both of our main results the aubin compactness theorem provides the underlying framework for the convergence to leray solutions