<Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their k _n -nearest neighbors, where {k _n : n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.      

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-estimation for the location parameters in stochastic volatility models

L ₁-estimation of a location parameter is studied for the “product type” stochastic volatility models. The asymptotic distribution of the L ₁-estimator is established under general conditions on the behavior of the distribution function of the errors near zero.     

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-norm packings from function fields

In this paper, we study some packings in a cube, namely, how to pack n points in a cube so as to maximize the minimal distance. The distance is induced by the L₁-norm which is analogous to the Hamming distance in coding theory. Two constructions with reasonable parameters are obtained, by using some results from a function field including divisor class group, narrow ray class group, and so on. We also present some asymptotic results of the two packings.     

Asymptotic analysis for loss probability of queues with finite <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 type structure

This paper discusses the asymptotic behavior of the loss probability for general queues with finite GI/M/1 type structure such as GI/M/c/K, SM/M/1/K and GI/MSP/1/K queues. We find an explicit expression for the asymptotic behavior of the loss probability as K tends to infinity. With the result, it is shown that the loss probability tends to 0 at a geometric rate. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> Extemal Polynomials Associated with Generalized Jacobi Weights

Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.     

Tensor products of symmetric functions over ℤ<Subscript>2</Subscript>

We calculate the homology and the cycles in tensor products of algebras of symmetric function over ℤ₂     

Generalized <Emphasis Type="Italic">T</Emphasis><Subscript>3</Subscript>-plot for testing high-dimensional normality

A new dimension-reduction graphical method for testing high-dimensional normality is developed by using the theory of spherical distributions and the idea of principal component analysis. The dimension reduction is realized by projecting high-dimensional data onto some selected eigenvector directions. The asymptotic statistical independence of the plotting functions on the selected eigenvector directions provides the principle for the new plot. A departure from multivariate normality of the raw data could be captured by at least one plot on the selected eigenvector direction. Acceptance regions associated with the plots are provided to enhance interpretability of the plots. Monte Carlo studies and an illustrative example show that the proposed graphical method has competitive power performance and improves the existing graphical method significantly in testing high-dimensional normality.     

On the asymptotically PT<Subscript>−1</Subscript> spaces

In this note, first, we show that the asymptotic subcone, the ultralimit, the completion of an asymptotically PT_-1 space are still an asymptotically PT_-1 space. Secondly, we consider two kinds of metric spaces, which have been considered by Ibragimov and Gromov, respectively. We show that they are asymptotically PT_-1 spaces under particular conditions, which provide some concrete examples of asymptotically PT_-1 spaces.     

A new multivariable <Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript><Emphasis Type="BoldItalic">ψ</Emphasis><Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript> summation formula

By multidimensional matrix inversion, combined with an A _r extension of Jackson’s ₈ φ ₇ summation formula by Milne, a new multivariable ₈ φ ₇ summation is derived. By a polynomial argument this ₈ φ ₇ summation is transformed to another multivariable ₈ φ ₇ summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating ₆ φ ₅ summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey’s very-well-poised ₆ ψ ₆ summation formula. Partly supported by FWF Austrian Science Fund grants P17563-N13, and S9607 (the second is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”).     

A fuzzy bootstrap test for the mean with <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">p,q</Emphasis></Subscript>-distance

In this paper, we consider the problem of testing a simple hypothesis about the mean of a fuzzy random variable. For this purpose, we take a distance between the sample mean and the mean in the null hypothesis as a test statistic. An asymptotic test about the fuzzy mean is obtained by using a central limit theorem. The asymptotical distribution is ω ₂-distribution. The ω ₂-distribution is only known for special cases, thus we have considered random LR-fuzzy numbers. In the fuzzy concept, in addition to the existence of several versions of the central limit theorem, there is another practical disadvantage: The limit law is, in most cases, difficult to handle. Therefore, the central limit theorem for fuzzy random variable does not seem to be a very useful tool to make inferences on the mean of fuzzy random variable. Thus we use the bootstrap technique. Finally, by means of a simulation study, we show that the bootstrap method is a powerful tool in the statistical hypothesis testing about the mean of fuzzy random variables.     

Localization of <Emphasis Type="Italic">LM</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-algebras

The aim of this paper is to define the localization LM _n -algebra of an LM _n —algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LM _n -algebra of fractions (defined in [3]) and the LM _n -algebra of fractions relative to an Λ—closed system (defined in Section 2) are LM _n -algebras of localization.     

Spin L-functions for <Emphasis Type="Italic">GSO</Emphasis><Subscript>10</Subscript> and <Emphasis Type="Italic">GSO</Emphasis><Subscript>12</Subscript>

Two multi-variable Rankin-Selberg integrals are studied. They may be regarded as extending the theory begun in [G-H1]. Each is shown to be Eulerian with the unramified contribution given explicitly in terms of partial Langlands L-functions.     

On asymptotic accuracy in <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> of a dynamical algorithm for reconstructing a disturbance

In the paper, a modification of the dynamical algorithm by Yu.S. Osipov and A.V. Kryazhimskii is suggested. This modification possesses in the space L ₁ an asymptotic order of accuracy arbitrarily close to 1/2. A possibility to attain this order in the class of finite-step dynamical algorithms is considered.     

Finite groups having the same prime graph as the group <Emphasis Type="Italic">A</Emphasis><Subscript>10</Subscript>

We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive. The infinitesimal matrix may contain a nilpotent part. In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have asymptotically homogeneous solutions in the critical case.     

A New Inequality for Weakly （K1, K2）-Quasiregular Mappings

We obtain a new inequality for weakly （K1,K2）-quasiregular mappings by using the McShane extension method. This inequality can be used to derive the self-improving regularity of （K1, K2）-Quasiregular Mappings.     

On generalized <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> and <Emphasis Type="Italic">R</Emphasis><Subscript>1</Subscript> spaces

Characterizations of some properties of generalized R ₀ and R ₁ topological spaces by using closure operator defined on a generalized topological space will be given. It is also shown that many results done in this area in some previous papers can be considered as special cases of our results.     

Sharp asymptotics of the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation error for interpolation on block partitions

Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in \mathbbR^d{\mathbb{R}^d} . We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases.     

Root-<Emphasis Type="Italic">n</Emphasis> consistency in weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-spaces for density estimators of invertible linear processes

The stationary density of an invertible linear processes can be estimated at the parametric rate by a convolution of residual-based kernel estimators. We have shown elsewhere that the convergence is uniform and that a functional central limit theorem holds in the space of continuous functions vanishing at infinity. Here we show that analogous results hold in weighted L ₁-spaces. We do not require smoothness of the innovation density.      

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.