Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces and Wiener amalgam spaces . We show that the periodic Brownian motion belongs locally in time to and for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that and form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space . Specifically, we prove that the Brownian motion belongs to for (s−1)p=−1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces , and indicate the endpoint large deviation estimates.     

Independence results for variants of sharply bounded induction

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊x/2⌋ but not ⌊x/2^y⌋), has recently been unconditionally separated from full bounded arithmetic S₂. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: (i) the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; (ii) extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at O∈R² for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

CERES in higher-order logic

We define a generalization of the first-order cut-elimination method CERES to higher-order logic. At the core of lies the computation of an (unsatisfiable) set of sequents (the characteristic sequent set) from a proof π of a sequent S. A refutation of in a higher-order resolution calculus can be used to transform cut-free parts of π (the proof projections) into a cut-free proof of S. An example illustrates the method and shows that can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.     

Sharp maximal inequality for nonnegative martingales

Let X be a nonnegative martingale, let H be a predictable process taking values in [−1,1] and let Y be an Itô integral of H with respect to X. We establish the bound and show that the constant 3 is the best possible.     

Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure.     

Some new results on multivariate dispersive ordering of generalized order statistics

Let be generalized order statistics based on a continuous distribution function F with parameters k and (m₁,…,m_n−1). Chen and Hu (2007) [8] investigated the sufficient conditions on F and on the parameters k and m_i’s such that , where , and is the Shaked-Shanthikumar multivariate dispersive order. Since the order does not possess the closure property under marginalization, one may naturally wonder whether the corresponding multivariate margins of the above random vectors are also ordered in the order . This is answered affirmatively in this paper. Some comparison results for generalized order statistics from two samples are presented. Potential applications are also mentioned.     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form and where p_j and P_j, j=0,…,n, are continuous functions defined on [a,b] and , is derived and then the Sturmian comparison theory for the corresponding 2nth-order equations l_α[x]=0 and L_α[y]=0 based on this identity is developed.