adaptive estimation of the dynamics of a discrete time stochastic volatility model abstract this paper is concerned with the discrete time stochastic volatility model y i exp x i 2 i x i 1 b x i x i i 1 where only y i is observed the model is rewritten as a particular hidden model z i x i e i x i 1 b x i x i i 1 where i and e i are independent sequences of i i d noise moreover the sequences x i and e i are independent and the distribution of e is known then our aim is to estimate the functions b and 2 when only observations z 1 z n are available we propose to estimate b f and b 2 2 f and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces by ratio strategy estimators of b and 2 are then deduced the mean square risk of the resulting estimators are studied and their rates are discussed lastly simulation experiments are provided constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples

optimal model selection for stationary data under various mixing conditions we build penalized least squares estimators of the marginal density of a stationary process using the slope algorithm and resampling penalties when the data are beta or tau mixing these estimators satisfy oracle inequalities with leading constant asymptotically equal to 1

data compression with low distortion and finite blocklength this paper considers lossy source coding of n dimensional memoryless sources and shows an explicit approximation to the minimum source coding rate required to sustain the probability of exceeding distortion d no greater than epsilon which is simpler than known dispersion based approximations our approach takes inspiration in the celebrated classical result stating that the shannon lower bound to rate distortion function becomes tight in the limit d to 0 we formulate an abstract version of the shannon lower bound that recovers both the classical shannon lower bound and the rate distortion function itself as special cases likewise we show that a nonasymptotic version of the abstract shannon lower bound recovers all previously known nonasymptotic converses a necessary and sufficient condition for the shannon lower bound to be attained exactly is presented it is demonstrated that whenever that condition is met the rate dispersion function is given simply by the varentropy of the source remarkably all finite alphabet sources with balanced distortion measures satisfy that condition in the range of low distortions most continuous sources violate that condition still we show that lattice quantizers closely approach the nonasymptotic shannon lower bound provided that the source density is smooth enough and the distortion is low this implies that fine multidimensional lattice coverings are nearly optimal in the rate distortion sense even at finite n the achievability proof technique is based on a new bound on the output entropy of lattice quantizers in terms of the differential entropy of the source the lattice cell size and a smoothness parameter of the source density the technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal

limiting spectral distribution of gram matrices associated with functionals of β mixing processes we give asymptotic spectral results for gram matrices of the form n 1 x n x t n where the entries of x n are dependent across both rows and columns and that are functionals of absolutely regular sequences and have only finite second moments we derive under mild dependence conditions in addition to an arithmetical decay condition on the mixing coefficients an integral equation of the stieltjes transform of the limiting spectral distribution of n 1 x n x t n in terms of the spectral density of the underlying process applications to examples of positive recurrent markov chains and dynamical systems are also given

low dimensional lattices ii subgroups of gl n ℤ the maximal finite irreducible groups of n x n integers for n 4 5 9 11 13 17 19 23 were determined by dade ryskov bulow plesken pohst and plesken as the automorphism groups of certain quadratic forms this paper presents a geometric description of the corresponding n dimensional lattices and gives coordinates which display their symmetries and minimal vectors some very interesting lattices appear

low dimensional lattices i quadratic forms of small determinant in this paper the first of a series dealing with low dimensional lattices and their applications we classify positive definite integral quadratic forms of determinant d 25 in dimensions up to a limit which ranges from 18 for d 1 to for d 25

moduli spaces of polarized symplectic o grady varieties and borcherds products we study moduli spaces of o grady s ten dimensional irreducible symplectic manifolds these moduli spaces are covers of modular varieties of dimension 21 namely quotients of hermitian symmetric domains by a suitable arithmetic group the interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group this makes it different from both the k3 and the k3 n case which are of dimension 19 and 20 respectively

asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization two results are given first using a result of csiszar 1973 the asymptotic i e high resolution low distortion performance for entropy constrained tessellating vector quantization heuristically derived by gersho 1979 is proven for all sources with finite differential entropy this implies using gersho s conjecture and zador s formula that tessellating vector quantizers are asymptotically optimal for this broad class of sources and generalizes a rigorous result of gish and pierce 1968 from the scalar to the vector case second the asymptotic performance is established for zamir and feder s 1992 randomized lattice quantization with the only assumption that the source has finite differential entropy it is proven that the low distortion performance of the zamir feder universal vector quantizer is asympotically the same as that of the deterministic lattice quantizer

rosenthal type inequalities for the maximum of partial sums of stationary processes and examples the aim of this paper is to propose new rosenthal type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations as in the recent results by peligrad et al proc amer math soc 135 2007 541 550 and rio j theoret probab 22 2009 146 163 the estimates of the moments are expressed in terms of the norms of projections of partial sums the proofs of the results are essentially based on a new maximal inequality generalizing the doob maximal inequality for martingales and dyadic induction various applications are also provided

low dimensional lattices iv the mass formula the mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them we restate the formula so as to make it easier to compute in particular we give a simple and reliable way to evaluate the 2 adic contribution our version unlike earlier ones is visibly invariant under scale changes and dualizing we use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series we also give tables of the standard mass the l series n m m s m odd and genera of lattices of determinant d 25

successive refinement of abstract sources in successive refinement of information the decoder refines its representation of the source progressively as it receives more encoded bits the rate distortion region of successive refinement describes the minimum rates required to attain the target distortions at each decoding stage in this paper we derive a parametric characterization of the rate distortion region for successive refinement of abstract sources our characterization extends csiszar s result to successive refinement and generalizes a result by tuncel and rose applicable for finite alphabet sources to abstract sources this characterization spawns a family of outer bounds to the rate distortion region it also enables an iterative algorithm for computing the rate distortion region which generalizes blahut s algorithm to successive refinement finally it leads a new nonasymptotic converse bound in all the scenarios where the dispersion is known this bound is second order optimal in our proof technique we avoid karush kuhn tucker conditions of optimality and we use basic tools of probability theory we leverage the donsker varadhan lemma for the minimization of relative entropy on abstract probability spaces

copula based time series with filtered nonstationarity abstract economic and financial time series data can exhibit nonstationary and nonlinear patterns simultaneously this paper studies copula based time series models that capture both patterns we introduce a procedure where nonstationarity is removed via a filtration and then the nonlinear temporal dependence in the filtered data is captured via a flexible markov copula we propose two estimators of the copula dependence parameters the parametric two step copula estimator where the marginal distribution of the filtered series is estimated parametrically and the semiparametric two step copula estimator where the marginal distribution is estimated via a rescaled empirical distribution of the filtered series we show that the limiting distribution of the parametric copula estimator depends on the nonstationary filtration and the parametric marginal distribution estimation and may be non normal surprisingly the limiting distribution of the semiparametric copula estimator using the filtered data is shown to be the same as that without nonstationary filtration which is normal and free of marginal distribution specification the simple and robust properties of the semiparametric copula estimators extend to models with misspecified copulas and facilitate statistical inferences such as hypothesis testing and model selection tests on semiparametric copula based dynamic models in the presence of nonstationarity monte carlo studies and real data applications are presented

local global principles for representations of quadratic forms we prove a local global principle for the problem of representations of quadratic forms by quadratic forms over in codimension 5 the proof uses the ergodic theory of p adic groups together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety

minimum positive entropy of complex enriques surface automorphisms we determine the minimum positive entropy of complex enriques surface automorphisms this together with mcmullen s work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of the enriques kodaira classification of complex surfaces

minimal vectors in the second exterior power of a lattice w x in c we studied the rankin k invariants which were defined by rankin w x in r as a natural generalization of the classical hermite invariant of a lattice in the following way let l be a lattice of maximal rank in an n dimensional euclidean space e we denote by d l the minimal k determinant of k dimensional sublattices of l and put g l s k k r n d l r det l the upper bound of g l for all lattices l in e is k k denoted by g n k it is readily verified that the k dimensional sublattice spanned by a k tuple of linearly independent vectors x x in l depends only on its 1 k image x n n x in the k th exterior power h l moreover h e 1 k inherits a canonical euclidean structure from the original scalar product on e which is defined for split multivectors that is vectors which can be written as x n n x by 1 k

the representation of binary quadratic forms by positive definite quaternary quadratic forms a quadratic z lattice l of rank n is defined to be k regular for a positive integer k n if l globally represents all quadratic z lattices of rank k which are represented everywhere locally by l it is shown that there exist only finitely many isometry classes of primitive positive definite quadratic z lattices of rank 4 which are 2 regular

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

low dimensional lattices iii perfect forms the perfect lattices or quadratic forms in dimensions n leqslant 7 found by barnes stacey and others are studied and their automorphism groups orbits of minimal vectors and eutactic coefficients are determined it is shown that just 30 of the 33 known seven dimensional perfect lattices are extreme a simple and self contained proof is given of the classification of perfect lattices in dimensions n leqslant 4 and hence of the determination of the densest lattice packings in these dimensions

low dimensional lattices vi voronoi reduction of three dimensional lattices the aim of this paper is to describe how the voronoi cell of a lattice changes as that lattice is continuously varied the usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice the present paper deals with dimensions n leq 3 a sequel will treat four dimensional lattices an elegant algorithm is given for the voronoi reduction of a three dimensional lattice leading to a new proof of voronoi s theorem that every lattice of dimension n leq 3 is of the first kind and of fedorov s classification of the three dimensional lattices into five types there is a very simple formula for the determinant of a three dimensional lattice in terms of its conorms

the shannon lower bound is asymptotically tight the shannon lower bound is one of the few lower bounds on the rate distortion function that holds for a large class of sources in this paper which considers exclusively norm based difference distortion measures it is demonstrated that its gap to the rate distortion function vanishes as the allowed distortion tends to zero for all sources having finite differential entropy and whose integer part has finite entropy conversely it is demonstrated that if the integer part of the source has infinite entropy then its rate distortion function is infinite for every finite distortion level thus the shannon lower bound provides an asymptotically tight bound on the rate distortion function if and only if the integer part of the source has finite entropy

a rigorous approach to high resolution entropy constrained vector quantization the nonnegativity of relative entropy implies that the differential entropy of a random vector x with probability density function pdf f is upper bounded by mathsf e left log g text x right for any arbitrary pdf g using this inequality with a cleverly chosen g we derive a lower bound on the asymptotic excess rate of entropy constrained vector quantization for d dimensional sources and r th power distortion where the asymptotic excess rate is defined as the difference between the smallest output entropy of a vector quantizer satisfying the distortion constraint and the rate distortion function in the limit as the distortion tends to zero specialized to the one dimensional case this lower bound coincides with the asymptotic excess rate achieved by a uniform quantizer thereby recovering the result by gish and pierce that uniform quantizers are asymptotically optimal as the allowed distortion tends to zero furthermore in the one dimensional case the derivation of the lower bound reveals a necessary condition for a sequence of quantizers to be asymptotically optimal this condition implies that any sequence of asymptotically optimal almost regular quantizers must converge to a uniform quantizer as the distortion tends to zero while the obtained lower bound itself is not novel to the best of our knowledge we present the first rigorous derivation that follows the direct approach by gish and pierce without resorting to heuristic high resolution approximations commonly found in the quantization literature furthermore our derivation holds for all d dimensional sources having finite differential entropy and whose integer part has finite entropy in contrast to gish and pierce we do not require additional constraints on the continuity or decay of the source pdf

positive definite n regular quadratic forms a positive definite integral quadratic form f is called n regular if f represents every quadratic form of rank n that is represented by the genus of f in this paper we show that for any integer n greater than or equal to 27 every n regular even form f is even n universal that is f represents all even respectively positive definite integral quadratic forms of rank n as an application we show that the minimal rank of n regular forms has an exponential lower bound for n as it increases

inequalities for absolutely regular sequences application to density estimation this paper investigates the problem of density estimation for absolutely regular observations in a first part we state two important results a new variance inequality and a rosenthal type inequality this allows us to study the p integrated risk p 2 of a large class of density estimators including kernel or projection estimators under the summability condition on the mixing coefficients k 0 k 1 p 2 k the rates obtained are those known to be optimal in the independent setting

fixed length lossy compression in the finite blocklength regime this paper studies the minimum achievable source coding rate as a function of blocklength n and probability that the distortion exceeds a given level d tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength for stationary memoryless sources with separable distortion the minimum rate achievable is shown to be closely approximated by r d v d n q 1 where r d is the rate distortion function v d is the rate dispersion a characteristic of the source which measures its stochastic variability and q 1 is the inverse of the standard gaussian complementary cumulative distribution function

finiteness theorems for positive definite n regular quadratic forms an integral quadratic form f of m variables is said to be n regular if f globally represents all quadratic forms of n variables that are represented by the genus of f for any n 2 it is shown that up to equivalence there are only finitely many primitive positive definite integral quadratic forms of n 3 variables that are n regular we also investigate similar finiteness results for almost n regular and spinor n regular quadratic forms it is shown that for any n 2 there are only finitely many equivalence classes of primitive positive definite spinor or almost n regular quadratic forms of n 2 variables these generalize the finiteness result for 2 regular quaternary quadratic forms proved by earnest 1994

on definite lattices bounded by integer surgeries along knots with slice genus at most 2 we classify the positive definite intersection forms that arise from smooth 4 manifolds with torsion free homology bounded by positive integer surgeries on the right handed trefoil a similar slightly less complete classification is given for the 2 5 torus knot and analogous results are obtained for integer surgeries on knots of slice genus at most two the proofs use input from yang mills instanton gauge theory and heegaard floer correction terms

low dimensional lattices v integral coordinates for integral lattices we say that an n dimensional classically integral lattice lambda is s integrable for an integer s if it can be described by vectors s frac 1 2 x 1 x k with all x i in mathbb z in a euclidean space of dimension k geqslant n equivalently lambda is s integrable if and only if any quadratic form f x corresponding to lambda can be written as s 1 times a sum of k squares of linear forms with integral coefficients or again if and only if the dual lattice lambda contains a eutactic star of scale s this paper gives many techniques for s integrating low dimensional lattices such as e 8 and the leech lattice a particular result is that any one dimensional lattice can be 1 integrated with k 4 this is lagrange s four squares theorem let phi s be the smallest dimension n in which there is an integral lattice that is not s integrable in 1937 ko and mordell showed that phi 1 6 we prove that phi 2 12 phi 3 14 21 leqslant phi 4 leqslant 25 16 leqslant phi 5 leqslant 22 phi s 4 leqslant 4s 2 s odd phi s leqslant 2 pi es 1 o 1 s even and phi s geqslant 2 ln ln s ln ln ln s 1 o 1

processus empiriques absolument reguliers et entropie universelle nous etendons le theoreme limite central fonctionnel de pollard pour les processus empiriques indexes par des familles de fonctions d entropie universelle integrable aux suites d observations absolument regulieres et stationnaires sous des conditions d entropie universelles sur raisonnables nous montrons un theoreme limite central uniforme pour les suites de variables aleatoires absolument regulieres et stationnaires sous la condition minimale de sommabilite des coefficients de melange n n

adaptive estimation in autoregression or mixing regression via model selection we study the problem of estimatingsome unknown regression function in a beta mixing dependent framework to this end we consider some collection of models which are finite dimensional spaces a penalized least squares estimator plse is built on a data driven selected model among this collection we state non asymptotic risk bounds for this plse and give several examples where the procedure can be applied autoregression regression with arithmetically beta mixing design points regression with mixing errors estimation in additive frameworks estimation of the order of the autoregression in addition we show that under a weak moment condition on the errors our estimator is adaptive in the minimax sense simultaneously over some family of besov balls

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

new dependence coefficients examples and applications to statistics to measure the dependence between a real valued random variable x and a algebra we consider four distances between the conditional distribution function of x given and the distribution function of x the coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations in particular we show that they are well adapted to functions of mixing sequences iterated random functions and dynamical systems starting from a new covariance inequality we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence we obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis under a minimal condition on the coefficients using recent results we show that our coefficients may be also used to obtain various exponential inequalities a concentration inequality for lipschitz functions and a berry esseen type inequality

the representation of quadratic forms by almost universal forms of higher rank in this article we prove that there are only finitely many positive definite integral quadratic forms of rank n 3 n 2 that represent all positive definite integral quadratic forms of rank n but finitely many exceptions furthermore we determine all diagonal quadratic forms having such property and its exceptions remaining four as candidates

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

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

representations of integral quadratic forms by sums of squares let g n be the minimum number of squares whose sum represents all positive definite integral quadratic forms of rank n which are represented by sums of squares in this article we first discuss representations of integers by unimodular lattices we then estimate the order of magnitude of the diameter of the 2 graph of unimodular lattices of rank n combining these results we prove g n o 3 n 2n log n we also provide a lower bound for g n finally we discuss s integrable lattices as an application of our method

wasserstein continuity of entropy and outer bounds for interference channels it is shown that under suitable regularity conditions differential entropy is o sqrt n lipschitz as a function of probability distributions on mathbb r n with respect to the quadratic wasserstein distance under similar conditions discrete shannon entropy is shown to be o n lipschitz in distributions over the product space with respect to ornstein s bar d distance wasserstein distance corresponding to the hamming distance these results together with talagrand s and marton s transportation information inequalities allow one to replace the unknown multi user interference with its independent identically distributed approximations as an application a new outer bound for the two user gaussian interference channel is proved which in particular settles the missing corner point problem of costa 1985