subelliptic spin c dirac operators ii basic estimates we assume that the manifold with boundary x has a spinc structure with spinor bundle s along the boundary this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is kahler in this case the spinc dirac operator agrees with along the boundary the induced cr structure on bx is integrable and either strictly pseudoconvex or strictly pseudoconcave we assume that e x is a complex vector bundle which has an infinite order integrable complex structure along bx compatible with that defined along bx in this paper we use boundary layer methods to prove subelliptic estimates for the twisted spinc dirac operator acting on sections on s e we use boundary conditions that are modifications of the classical neumann condition these results are proved by using the extended heisenberg calculus

invariance principles for partial sum processes and empirical processes indexed by sets let x j j q be independent identically formed random elements with values in a banach spaceb indexed byq tuples of positive integers we obtain the almost sure approximation as well as the approximation in probability of the partial sum process sum limits j in na x j a in a by a partial sum process sum limits j in na y j a in a asn uniformly over all setsa in a certain class a of subsets of theq dimensional unit cube here y j j q are independent identically distributed gaussian random variables these results are then applied to obtain the approximation of empirical processes over sets and indexed by sets by gaussian partial sum processes indexed by sets

morse inequalities for fourier components of kohn rossi cohomology of cr manifolds with s 1 action let x be a compact connected cr manifold of dimension 2n 1 n ge 2 with a transversal cr s 1 action on x we study the fourier components of the kohn rossi cohomology with respect to the s 1 action by studying the szego kernel of the fourier components we establish the morse inequalities on x using the morse inequalities we have established on x we prove that there are abundant cr functions on x when x is weakly pseudoconvex and strongly pseudoconvex at a point

fourier integral operators on lie groupoids abstract as announced in 36 we develop a calculus of fourier integral g operators on any lie groupoid g for that purpose we study lagrangian conic submanifolds of the symplectic groupoid t g this includes their product transposition and inversion we also study the relationship between these lagrangian submanifolds and the equivariant families of lagrangian submanifolds of t g x t g x parametrized by the units x g 0 of g this allows us to select a subclass of lagrangian distributions on any lie groupoid g that deserve the name of fourier integral g operators g fios by construction the class of g fios contains the class of equivariant families of ordinary fourier integral operators on the manifolds g x x g 0 we then develop for g fios the first stages of the calculus in the spirit of hormander s work finally we illustrate this calculus in the case of manifolds with boundary

strongly pseudoconvex cr structures over small balls part i an a priori estimate the purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex cr structures over small balls along the line developed by d c spencer c b morrey j j kohn and l nirenberg 5 6 7 8 for strongly pseudoconvex complex manifolds we consider a strongly pseudoconvex cr structure on a manifold of real dimension 2n 1 in part i we establish the a priori estimate for the neumann boundary value problem on the complex associated with the structure in the case the structure is induced by an embedding in c and restricted to a small ball of special type provided i c q c n 3 where q is the degree of differential forms we consider in part ii we derive the regularity theorem of solutions of the neumann boundary value problem based on the a priori estimate of part i as an application we prove in part il that when n 5 the structure is realized on a neighborhood of a reference point by an embedding in cn we note here that l nirenberg gave an example of a strongly pseudoconvex cr structure which cannot be induced by such an embedding when n 2 9 l boutet de monvel proved that in the case of such structures over a compact manifold we can find such an embedding when n 3 1 there are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and cr structures firstly the principal parts of the associated laplacians in cr structures are no longer elliptic secondly we have characteristic points on the boundary as a result when we follow over the predecessor s calculation in our case we end up with terms which may not be absorbed in the main non negative terms fortunately these leftover terms depend mostly on the levi form of the cr structure and the hessian form with respect to the cr structure of the equation of the boundary of our

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

non realizable cr structures the examples given by nirenberg in 5 and 6 show that not all three dimensional cr manifolds may be realized as real hypersurfaces in 1172 these examples are perturbations of the cr structure on the sphere in this paper we extend his construction and show that any strictly pseudo convex realizable cr structure of dimension 3 may be perturbed so as to obtain a non realizable one this is done in sect 1 in sect 2 we show non realizability for higher dimensional structures provided the levi form is non degenerate and has only one positive eigenvalue in our last section we summarize cartan s approach to cr geometry from this viewpoint it was important to decide if nirenberg s example was really limited to perturbations of the sphere we have made this paper self contained and have provided proofs of several known results for instance lemma 1 l and the first part of lemma 1 5 were proved by lewy 2 and lemma 1 2 is the well known observation that a strictly pseudo convex hypersurface is convex in some coordinate system

stability of the embeddability under perturbations of the cr structure for compact cr manifolds we study the stability of the embeddability of compact 2 concave cr manifolds in complex manifolds under small horizontal perturbations of the cr structure

on the stability of equivariant embedding of compact cr manifolds with circle action we prove the stability of the equivariant embedding of compact strictly pseudoconvex cr manifolds with transversal cr circle action under circle invariant deformations of the cr structures

hellinger consistency of certain nonparametric maximum likelihood estimators consider a class mathscr p p theta theta in theta of probability measures on a measurable space mathscr x mathscr a dominated by a sigma finite measure mu let f theta dp theta d mu theta in theta and let theta n be a maximum likelihood estimator based on n independent observations from p theta 0 theta 0 in theta we use results from empirical process theory to obtain convergence for the hellinger distance h f hat theta n f theta 0 under certain entropy conditions on the class of densities f theta theta in theta the examples we present are a model with interval censored observations smooth densities monotone densities and convolution models in most examples the convexity of the class of densities is of special importance

cr structures with group action and extendability of cr functions part i presents results on local embedding of cr structures we consider an abstract cr manifold whose structure is invariant under a transversal lie group action we show that such a manifold can always be locally embedded in complex space as a generic submanifold the proof is based on selection of canonical coordinates and repeated use of the newlander nirenberg theorem 13 when the lie group is abelian the embedding can be given a particularly simple form let l 1 be the codimension of our submanifold called m throughout the paper it is then convenient to denote by n i the dimension of the ambient complex space and by zl z w 1 w t the complex coordinates we shall systematically write

subelliptic boundary conditions for spinc dirac operators gluing relative indices and tame fredholm pairs abstract let x be a spin manifold with boundary such that the spin structure is defined near the boundary by an almost complex structure which is either strictly pseudoconvex or pseudoconcave and hence contact using generalized szeg projectors we define modified neumann boundary conditions eo for spinors which lead to subelliptic fredholm boundary value problems for the spin dirac operator eo to study the index of these boundary value problems we introduce a generalization of fredholm pairs to the tame category in this context we show that the index of the graph closure of eo eo equals the tame relative index on the boundary between eo and the calderon projector let x0 and x1 be strictly pseudoconvex spin manifolds as above let bx1 bx0 be a contact diffeomorphism s0 s1 denote generalized szeg projectors on bx0 bx1 respectively and 0eo 1eo the subelliptic boundary conditions they define if x1 is the manifold x1 with its orientation reversed then the glued manifold x x0 x1 has a canonical spin structure and dirac operator xeo applying these results we obtain a formula for the relative index r ind s0 s1 as a special case this formula verifies a conjecture of atiyah and weinstein 1997 rims kokyuroku 1014 1 14 for the index of the quantization of a contact transformation between cosphere bundles

the law of large numbers for partial sum processes indexed by sets on donne des conditions necessaires et suffisantes pour la loi des grands nombres pour des processus de sommes partielles indicees par des ensembles de 0 1 d et basees sur des variables aleatoires integrables i i d

a uniform central limit theorem for set indexed partial sum processes with finite variance on obtient des resultats de convergence faible qui ameliorent ceux de pyke 1983 et de bass et pyke 1984

exponential inequalities for martingales with application to maximum likelihood estimation for counting processes we obtain an exponential probability inequality for martingales and a uniform probability inequality for the process gdn where n is a counting process and where g varies within a class of predictable functions g for the latter we use techniques from empirical process theory the uniform inequality is shown to hold under certain entropy conditions on g as an application we consider rates of convergence for nonparametric maximum likelihood estimators for counting processes a similar result for discrete time observations is also presented

inverse moment bounds for sample autocovariance matrices based on detrended time series and their applications abstract in this paper we assume that observations are generated by a linear regression model with short or long memory dependent errors we establish inverse moment bounds for k n dimensional sample autocovariance matrices based on the least squares residuals also known as the detrended time series where k n n k n and n is the sample size these results are then used to derive the mean square error bounds for the finite predictor coefficients of the underlying error process based on the detrended time series we further estimate the inverse of the n dimensional autocovariance matrix r n 1 of the error process using the banded cholesky factorization by making use of the aforementioned inverse moment bounds we obtain the convergence of moments of the difference between the proposed estimator and r n 1 under spectral norm

subelliptic mathrm spin _ mathbb c dirac operators i let x be a compact kahler manifold with strictly pseudoconvex bound ary y in this setting the spinc dirac operator is canonically identified with c x 0 e c x 0 o we consider modifications of the classi cal neumann conditions that define fredholm problems for the spinc dirac operator in part 2 7 we use boundary layer methods to obtain subelliptic estimates for these boundary value problems using these results we obtain an expression for the finite part of the holomorphic euler characteristic of a strictly pseudoconvex manifold as the index of a spinc dirac operator with a subellip tic boundary condition we also prove an analogue of the agranovich dynin formula expressing the change in the index in terms of a relative index on the boundary if x is a complex manifold partitioned by a strictly pseudoconvex hypersurface then we obtain formulae for the holomorphic euler characteristic of x as sums of indices of spinc dirac operators on the components this is a subelliptic analogue of bojarski s formula in the elliptic case

the central limit theorem for empirical processes on vapnik cervonenkis classes sufficient conditions which under weak additional assumptions are also necessary are found for the central limit theorem for empirical processes indexed by classes of functions which have the vapnik cervonenkis property this improves an earlier theorem of pollard 1982a and leads to necessary and sufficient conditions for the clt for weighted empirical processes indexed by vapnik cervonenkis classes of sets

convergence of gaussian quasi likelihood random fields for ergodic levy driven sde observed at high frequency this paper investigates the gaussian quasi likelihood estimation of an exponentially ergodic multidimensional markov process which is expressed as a solution to a levy driven stochastic differential equations whose coefficients are supposed to be known except for the finite dimensional parameters to be estimated we suppose that the process is observed under the condition for the rapidly increasing experimental design by means of the polynomial type large deviation inequality the mighty convergence of the corresponding statistical random fields is derived which especially leads to the asymptotic normality at rate p nhn for all the target parameters and also to the convergence of their moments in our results the diffusion coefficient may be degenerate or even null although the resulting estimator is not asymptotically efficient in the presence of jumps we do not require any specific form of the driving levy measure rendering that the proposed estimation procedure is practical and somewhat robust to underlying model specification

law of the iterated logarithm for set indexed partial sum processes with finite variance is relatively compact in the uniform topology and the set of limit points is the set of functions which are absolutely continuous and whose derivatives have l 2 norm less than or equal to 1 using skorokhod embedding strassen showed that random walks also obeyed a functional lil and that the usual lil as well as many other interesting results could be obtained as a consequence the extension of strassen s results to brownian motion indexed by points in and to partial sum processes indexed by points in ir was done by pyke 7 and wichura 9 respectively for a brownian motion indexed by a large collection d of sets the functional law was proved by bass and pyke 2 and then a type of skorokhod embedding was used to extend this for partial sum processes however if h 5 is the log entropy of d defined in sect 2 it was necessary that h 6 0 moreover the sets in d had to have uniformly smooth boundaries morrow and philipp 6j were able to wea

subelliptic mathrm spin _ mathbb c dirac operators iii the atiyah weinstein conjecture in this paper we extend the results obtained in 9 10 to manifolds with spin c structures defined near the boundary by an almost complex structure we show that on such a manifold with a strictly pseudoconvex boundary there are modified neumann boundary conditions defined by projection operators r eo which give subelliptic fredholm problems for the spin c dirac operator eo we introduce a generalization of fredholm pairs to the tame category in this context we show that the index of the graph closure of eo r eo equals

empirical risk minimization in inverse problems we study estimation of a multivariate function f r d r when the observations are available from the function af where a is a known linear operator both the gaussian white noise model and density estimation are studied we define an l 2 empirical risk functional which is used to define a net minimizer and a dense empirical risk minimizer upper bounds for the mean integrated squared error of the estimators are given the upper bounds show how the difficulty of the estimation depends on the operator through the norm of the adjoint of the inverse of the operator and on the underlying function class through the entropy of the class corresponding lower bounds are also derived as examples we consider convolution operators and the radon transform in these examples the estimators achieve the optimal rates of convergence furthermore a new type of oracle inequality is given for inverse problems in additive models

on global representation of lagrangian distributions and solutions of hyperbolic equations in this paper we develop a new approach to the theory of fourier integral operators it allows us to represent the schwartz kernel of a fourier integral operator by one oscillatory integral with a complex phase function we consider fourier integral operators associated with canonical transformations having in mind applications to hyperbolic equations as a by product we obtain yet another formula for the maslov index c 1994 john wiley sons inc

a large deviation result for parameter estimators and its application to nonlinear regression analysis elaborating on the work of ibragimov and has minskii 1981 we prove a law of large deviations lld for m estimators i e those estimators which maximize a functional continuous in the parameter of the observations this lld is applied using the results of petrov 1975 to the problem of parametrical nonlinear regression in the situation of discrete time independent errors and regression functions which are continuous in the parameter this improves a result of prakasa rao 1984

geometry of pseudodifferential algebra bundles and fourier integral operators we study the geometry and topology of filtered algebra bundles z over a smooth manifold x with typical fiber z z v the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle v over the compact manifold z and of integral order first a theorem of duistermaat and singer is generalized to the assertion that the group of projective invertible fourier integral operators pg fc z v is precisely the automorphism group of the filtered algebra of pseudodifferential operators we replace some of the arguments in their work by microlocal ones thereby removing the topological assumption we define a natural class of connections and b fields on the principal bundle to which z is associated and obtain a de rham representative of the dixmier douady class in terms of the outer derivation on the lie algebra and the residue trace of guillemin and wodzicki the resulting formula only depends on the formal symbol algebra z examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fiber bundle over x

fourier integral operators and the index of symplectomorphisms on manifolds with boundary given two compact manifolds with boundary x y and a boundary preserving symplectomorphism t y 0 t x 0 which is one homogeneous in the fibers and satisfies the transmission condition we introduce fourier integral operators of boutet de monvel type associated with we study their mapping properties between sobolev spaces develop a calculus and prove a egorov type theorem we also introduce a notion of ellipticity which implies the fredholm property finally we show how in the spirit of a classical construction by a weinstein a fredholm operator of this type can be associated with and a section of the maslov bundle if dim y 2 or the maslov bundle is trivial the index is independent of the section and thus an invariant of the symplectomorphism

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix

delta method in large deviations and moderate deviations for estimators the delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities the large and moderate deviation theory can achieve this goal motivated by the delta method in weak convergence a general delta method in large deviations is proposed the new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the wilcoxon statistic the kaplan meier estimator the empirical quantile processes and the empirical copula function we also improve the existing moderate deviations results for m estimators and l statistics by the new method some applications of moderate deviations to statistical hypothesis testing are provided

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

a bernstein type inequality and moderate deviations for weakly dependent sequences in this paper we present a bernstein type tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that is not necessarily bounded the class considered includes geometrically and subgeometrically strongly mixing sequences the result is then used to derive asymptotic moderate deviation results applications are given for classes of markov chains iterated lipschitz models and functions of linear processes with absolutely regular innovations

variance of set indexed sums of mixing random variables and weak convergence of set indexed processes on obtient une borne uniforme pour la variance d un processus de somme partielle indice sur un ensemble sous une condition de melange

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n