some properties of schur complements and diagonal schur complements of diagonally dominant matrices in this paper we prove that the diagonal schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix the same holds for the diagonal schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular h matrix we point out that under certain assumptions the diagonal schur complement of a strictly doubly doubly product diagonally dominant matrix is also strictly doubly doubly product diagonally dominant further we provide the distribution of the real parts of eigenvalues of a diagonal schur complement of h matrix we also show that the schur complement of a diagonally dominant matrix is not always diagonally dominant by a numerical example and then obtain a sufficient condition to ensure that the schur complement of a diagonally dominant matrix is diagonally dominant

a note on partitioned determinants a formula for the determinant of a partitioned matrix possibly with singular submatrices is derived and applied to some psychometric and numerical problems

additive functionals of several time reversible markov processes i 1 additive functionals of markov processes have been studied by many authors we refer to 1 and 5 for the early history and to 161 i1719 61 181 f or more recent developments the most complete results for the case of a time reversible process are due to fukushima lo in this paper we study additive functionals of several independent time reversible markov processes the results are useful in particular for investigating a certain class of gaussian random fields which includes the free field of euclidean quantum field theory and the so called brownian sheet 9 our approach is different from that of fukushima even in the case of one markov process his results do not imply ours in the introduction first additive functionals of brownian motions are discussed on an intuitive level then the general concept of standard timereversible markov processes is outlined and the rigorous definition of additive functionals is given on this base the principal results of the paper are formulated we mention also some open problems

some dimension results for super brownian motion the dawson watanabe super brownian motion has been intensively studied in the last few years in particular there has been much work concerning the hausdorff dimension of certain remarkable sets related to super brownian motion we contribute to this study in the following way let yt t 0 be a super brownian motion on d d 2 andh be a borel subset of d we determine the hausdorff dimension of t 0 suppyt h o improving and generalizing a result of krone we also obtain a new proof of a result of tribe which gives whend 4 the hausdorff dimension of cup t in rm b suppy t as a function of the dimension ofb

fisher information prior probabilities and the state determination of spin 1 2 and spin 1 systems the fisher information matrix i2p for a p p density matrix is taken to be the 2p2 p dimensional information matrix of a 2p variate real normal distribution with an a priori zero mean vector and a covariance matrix 2p the two diagonal blocks of which are formed from the symmetric real component of and the two off diagonal blocks from the skew symmetric imaginary part of the inverse square root ni2p 1 2 where n is the number of observations serves as a unitary invariant prior distribution over the precision matrices 2p 1 and ni2p 1 2 over 2p and consequently over the p p mixed and pure density matrices the availability of these priors should facilitate the process of state determination an uncertainty principle for the estimation of is formulated the determinant of 2p proves for integral p 1 to equal 2 p 2

alternative characterizations of the extended wedderburn guttman theorem let a be a u by v matrix of rank a and let m and n be u by p and v by q matrices respectively where p is not necessarily equal to q or rank m an min p q takane and yanai y takane h yanai on the wedderburn guttman theorem linear algebra appl 410 2005 267 278 investigated the conditions under which rank a an m an m a rank a rank an m an m a this is called the extended wedderburn guttman theorem in this paper we give alternative characterizations of these conditions using the product singular value decomposition psvd of matrix triplets

estimation and testing when explanatory variables are endogenous an application to a demand system abstract in a system of linear equations where some of the right hand side explanatory variables are correlated with the equation errors a priori information in the form of linear restrictions is used to transform the model into a system where all right hand variables are exogenous the resulting equations can be estimated using standard multivariate methods it is shown how the simple errors in variables model places testable restrictions on some of the parameters of the model the methods are applied to a system of demand equations using quarterly data for the u k

two step two stage least squares estimation in models with rational expectations abstract the paper develops a two step estimator for use in rational expectations models with autocorrelated residuals and predetermined but not strictly exogenous instruments the estimator extends the applicability of mccallum s 1976 error in variablesapproach to estimating such models and is asymptotically efficient in a class of intrumental variables estimators as an application we use instrumental variables techniques to estimate taylor s 1979 rational expectations macroeconomic model of the united states

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

multifractal decompositions of moran fractals we present a rigorous construction and generalization of the multifractal decomposition for moran fractals with infinite product measure the generalization is specified by a system of nonnegative weights in the partition sum all the usual smooth properties of the f theory are recovered for the case that the weights are equal to unity the generalized spectrum f w is invariant to a group of gauge transformations of the weights and in addition need no longer be concave in case the fractal is a cantor set generated by an iterated function system of similarities is the pointwise dimension of the measure we discuss properties of some examples

absolute continuity results for superprocesses with some applications let x1 and x2 be instances of a measure valued dawsonwatanabe 4 super process where the underlying spatial motions are given by a borel right process x and where the branching mechanism has finite variance a necessary and sufficient condition on x0 and x2 is found for the law of xi 2 to be absolutely continuous with respect to the law of x2 the conditions are the natural absolute continuity conditions on s but some care must be taken with the set of times s t being considered the result is used to study the closed support of super brownian motion and give sufficient conditions for the existence of a nontrivial collision measure for a pair of independent superlevy processes or more generally for a super levy process and a fixed measure the collision measure gauges the extent of overlap of the two measures as a final application we give an elementary proof of the instantaneous propagation of a super levy process to all points to which the underlying levy process can jump this result is then extended to a much larger class of superprocesses using different techniques

properties of the matrix a xy abstract the main topic of this paper is the matrix v a xy where a is a nonsingular complex k k matrix and x and y are k p complex matrices of full column rank because properties of the matrix v can be derived from those of the matrix q i xy we will consider in particular the case where a i for the case that y x i so that q is singular we will derive the moore penrose inverse of q the moore penrose inverse of v in case y a 1 x i then easily follows finally we will focus on the eigenvalues and eigenvectors of the real matrix d xy with d diagonal

a uniform law of large numbers for dependent and heterogeneous data processes uniform laws of large numbers ullns consider sums of the form n 1 t n 1 q t z t eq t z t where z t denotes a stochastic data generating process that takes its values in a space z is an element of the parameter space and q t z r ullns provide conditions under which the above sum converges to zero uniformly over the parameter space the purpose of the present note is to introduce a new generic ulln it maintains a set of assumptions that is relatively easy to verify and allows at the same time the analysis of a wide variety of estimators and models of interest in economics

the positive semidefiniteness of partitioned matrices abstract the positive semidefiniteness of a partitioned matrix is characterized in terms of its submatrices the result is applied to a variety of problems concerning lowner ordered matrices which need not be partitioned themselves

proxies versus omitted variables in regression analysis when some variables to be used in a regression analysis contain measurement error the ols estimator will be inconsistent when these variables are left out ols will also be inconsistent in an appropriate metric the first type of inconsistency is smaller than the second type

on the inverse of a special schur complement abstract the inverse of a schur complement is a very useful tool in many algorithms for the computation of the matrix inversion in this paper we study the inverse of a special schur complement cd 1 b we proved that always exist some x and y such that xdy is a inverse of cd 1 b furthermore using minimal rank properties we give some explicit expressions for x y and the inverse of cd 1 b numerical examples also given

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

a multifractal formalism let x be a metric space and a borel probability measure on x for q t r and e x write formula then hq t and pq t are borel measures hq t is a multifractal generalization of the centered hausdorff measure and pq t is a multifractal generalization of the packing measure the measures hq t and pq t define for a fixed q in the usual way a generalized hausdorff dimension dimq e and a generalized packing dimension dimq e of subsets e of x we study the functions b q dimq supp b q dimq supp and their relation to the so called multifractal spectra functions of formula we prove among other things that f f is bounded from above by the legendre transform of b b and that equality holds for graph directed self similar measures and cookie cutter measures finally we discuss the connection with generalized renyi dimensions

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

a schur complement approach for computing subcovariance matrices arising in a road safety measure modelling this paper deals with the determination of the analytical expression of a real partitioned matrix inverse a formal inversion method is suggested through schur complements and the elements of any inverse matrix block are explained irrespective of the size of the considered matrix the suggested approach does not need any matrix inversion numerical program this methodology is applied to a fisher information matrix relating to a multidimensional modelling of road accident data when a road safety measure is applied on different experiment sites an example of formal calculation and interpretation is given to support our approach

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

the structure of us food demand abstract an exactly aggregable system of gorman engel curves for us food consumption is developed and implemented box cox transformations on prices and income nest functional form the model nests rank up to rank three the model is estimated by nonlinear three stage least squares with annual time series data on 21 foods 17 nutrients age and race demographics and the distribution of income for 1919 1941 and 1947 2000 results are consistent with full rank three point estimates for the box cox parameters on income and prices are 0 86 and 1 09 respectively strongly rejecting zero and one in both cases no statistical evidence of serial correlation specification errors or parameter instability is found

differentiability of l q spectrum and multifractal decomposition by using infinite graph directed ifss abstract by constructing an infinite graph directed iterated function system associated with a finite iterated function system we develop a new approach for proving the differentiability of the l q spectrum and establishing the multifractal formalism of certain self similar measures with overlaps especially those defined by similitudes with different contraction ratios we apply our technique to a well known class of self similar measures of generalized finite type

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

homogeneity and endogeneity in systems of demand equations abstract in estimating systems of demand equations one of the right hand side explanatory variables expenditure may be endogenous in the sense that it is correlated with the equation error if the assumption of homogeneity of degree zero in prices and nominal income is imposed on the system it turns out it is still possible to estimate the parameters of the system even when expenditure is endogenous the estimation procedure is simple requiring just one additional ordinary least squares regression the paper also demostrates that a model in which homogeneity is tested with expenditure assumed exogenous is exactly equivalent to a model in which the exogeneity of expenditure is tested with homogeneity imposed previous tests of demand systems which have rejected the homogeneity postulate might therefore be reinterpreted instead as rejecting the hypothesis of exogeneity of expenditure with homogeneity of degree zero in prices and nominal income taken as given

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

on estimation and testing when explanatory variables are partly endogenous abstract when some of explanatory variables are correlated with regression errors the standard econometric theory advocates the use of instrumental variables estimators this paper considers a special case where applied researchers do not have a sufficient number of instruments but instead they do have prior information on parameters in the form of parameter restrictions the paper proposes a simple method of estimation and hypothesis testing in the above context the results are extended to restricted gmm estimation in nonlinear context

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

on existence and uniqueness of stationary points in semi infinite optimization the paper deals with semi infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints we generalize the concept of strongly stable stationary points which was introduced by kojima for finite problems it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem where perturbations up to second order are allowed under the extended mangasarian fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions in particular we discuss the case where the reduction approach is not satisfied

perturbation and interlace theorems for the unitary eigenvalue problem abstract two aspects of the perturbation problem for the eigenvalues of a unitary matrix u are treated firstly analogues of the hoffman wielandt theorem and a weyl type theorem proved by bhatia and davis are derived which are based on a different measure of the distance of spectra using a suitable parametrization of the unit circle by an angle the new results are called tangent theorems in contrast to the first mentioned well known results which are sine theorems moreover we illuminate the unknown minimizing permutations in the above weyl type theorems with respect to their angles the eigenvalues of u and the perturbed matrix are naturally ordered on the unit circle counterclockwise after a point is cut on the unit circle we prove a well known open conjecture there exists a cutting point such that the weyl type theorems both sine and tangent are true when the ordered eigenvalues of u and are paired with each other secondly the cauchy interlacing theorem for hermitian matrices is generalized it is shown that certain modified principal submatrices of u called the modified kth leading principal submatrices have the property that their eigenvalues interlace those of u finally we discuss block reflectors appearing in the description of the modified principal submatrices and generalize a result of schreiber and parlett

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

multifractal decompositions of digraph recursive fractals we prove that the multifractal decomposition behaves as expected for a family of sets k known as digraph recursive fractals using measures of markov type for each value of a parameter between a minimum min and maximum max we define multifractal components k of k and show that they are fractals in the sense of taylor the dimension f of k is computed from the data of the problem the typical concave multifractal f dimension spectrum curve results

schur complements and matrix inequalities in the lowner ordering abstract the purpose of this paper is to present a matrix inequality on the kronecker product that unifies the proofs of many existing matrix inequalities in the lowner partial ordering on the sum ordinary product and hadamard schur product schur complements serve as the basic tool

a generalized finite type condition for iterated function systems abstract we study iterated function systems ifss of contractive similitudes on r d with overlaps we introduce a generalized finite type condition which extends a more restrictive condition in s m ngai y wang hausdorff dimension of self similar sets with overlaps j london math soc 2 63 3 2001 655 672 and allows us to include some ifss of contractive similitudes whose contraction ratios are not exponentially commensurable we show that the generalized finite type condition implies the weak separation property under this condition we can identify the attractor of the ifs with that of a graph directed ifs and by modifying a setup of mauldin and williams r d mauldin s c williams hausdorff dimension in graph directed constructions trans amer math soc 309 1988 811 829 we can compute the hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix

thick points of super brownian motion we determine for a super brownian motion xt t 0 in d d 3 the precise gauge function such that almost surely on survival up to time t open image in new window improving a result of barlow evans and perkins about the most visited sites of super brownian motion we also determine upper and lower bounds for the hausdorff dimension spectrum of thick points refining the multifractal analysis of super brownian motion by taylor and perkins the upper bound conjectured to be sharp involves a constant which can be characterized in terms of the upper tails of the associated equilibrium palm distribution