Association and random measures

Summary Our point of departure is the result, due to Burton and Waymire, that every infinitely divisible random measure has the property variously known as association, positive correlations, or the FKG property. This leads into a study of stationary, associated random measures onR ^d . We establish simple necessary and sufficient conditions for ergodicity and mixing when second moments are present. We also study the second moment condition that is usually referrent to as finite susceptibility. As one consequence of these results, we can easily rederive some central limit theorems of Burton and Waymire. Using association techniques, we obtain a law of the iterated logarithm for infinitely divisible random measures under simple moment hypotheses. Finally, we apply these results to a class of stationary random measures related to measure-valued Markov branching processes.Research supported in part by NSF Grant DMS-8701212 at the University of Virginia     

Monte Carlo approximate tensor moment simulations

An algorithm to generate samples with approximate first‐order, second‐order, third‐order, and fourth‐order moments is presented by extending the Cholesky matrix decomposition to a Cholesky tensor decomposition of an arbitrary order. The tensor decomposition of the first‐order, second‐order, third‐order, and fourth‐order objective moments generates a non‐linear system of equations. The algorithm solves these equations by numerical methods. The results show that the optimization algorithm delivers samples with an approximate residual error of less than 10¹⁶ between the components of the objective and the sample moments. The algorithm is extended for a n‐th‐order approximate tensor moment version, and simulations of non‐normal samples replicated from distributions with asymmetries and heavy tails are presented. An application for sensitivity analysis of portfolio risk assessment with Value‐at‐Risk (VaR) is provided. A comparison with previous methods available in the literature suggests that the methodology proposed reduces the error of the objective moments in the generated samples. ? ? JEL Classification: C14, C15, G32.
Copyright © 2016 John Wiley & Sons, Ltd.     

An asymptotic approach for a semi‐Markovian inventory model of type (s,S)

In this study, we constructed a stochastic process (X(t)) that expresses a semi‐Markovian inventory model of type (s, S) and it is shown that this process is ergodic under some weak conditions. Moreover, we obtained exact and asymptotic expressions for the n^th order moments (n = 1,2,3, … ) of ergodic distribution of the process X(t), as S ? s → ∞ . Finally, we tested how close the obtained approximation formulas are to the exact expressions. Copyright © 2012 John Wiley & Sons, Ltd.     

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.     

An investigation of phase-distribution moment-matching algorithms for use in queueing models

Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.     

A Study of Probability Measures Through Commutators

The first-order moments and two families of commutators are proven to determine uniquely the moments of a probability measure on ℝ ^d . These families are the commutators between the annihilation and creation operators, and the commutators between the annihilation and preservation operators. An explicit method for recovering the moments from these commutators and first-order moments is presented.      

Commutativity for a Certain Class of Rings

We first establish the commutativity for the semiprime ring satisfying [x ⁿ , y]x ^r = ±y ^s[x, y ^m]y ^t for all x, y in R, where m, n, r, s, and t are fixed non-negative integers, and further, we investigate the commutativity of rings with unity under some additional hypotheses. Moreover, it is also shown that the above result is true for s-unital rings. Also, we provide some counterexamples which show that the hypotheses of our theorems are not altogether superfluous. The results of this paper generalize some of the well-known commutativity theorems for rings which are right s-unital.     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

Anisotropic Meshless Frames on ℝ n

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝ ⁿ . The wavelets we construct are C ^∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C ^∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.     

The moments of the extended compound point process with applications

The Concept of Compound point(or counting)process is generalized to account for situations in which the compound events are described by a stochastic process rather than a random variable. Quasi–closed form expressions for the mean and the variance of this extended compounded point process are obtained in terms of the first two moments of the underlying point process and of the process describing the compound events. An expression for the higher moments is also obtained. The results are applied to obtain the mean and variance of the number of busy channels in a pp^x/G/∞ queue. (PP=point process) which generalises results of the M(t)^x/G/∞ and GI^x/G/∞ queue. Other applications include the derivation of the first two moments of the total backlog and of the revenues from telephone traffic in a PP/G/∞ queue. Finally, a special empirical model is considered     

Existence Theorems for Periodic Differential Inclusions in IR^N

We study the periodic problem for differential inclusions in R~N.First we look for extremal periodicsolutions.Using techniques from multivalued analysis and a fixed point argument we establish an existencetheorem under some general hypotheses.We also consider the“nonconvex periodic problem”under lowersemicontinuity hypotheses,and the“convex periodic problem”under general upper semicontinuity hypotheseson the multivalued vector field.For both problems,we prove existence theorems under very general hypotheses.Our approach extends existing results in the literature and appear to be the most general results on the nonconvexperiodic problem.     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Hierarchies of higher order kernels

Summary Recent literature on functional estimation has shown the importance of kernels with vanishing moments although no general framework was given to build kernels of increasing order apart from some specific methods based on moment relationships. The purpose of the present paper is to develop such a framework and to show how to build higher order kernels with nice properties and to solve optimization problems about kernels. The proofs given here, unlike standard variational arguments, explain why some hierarchies of kernels do have optimality properties. Applications are given to functional estimation in a general context. In the last section special attention is paid to density estimates based on kernels of order (m, r), i.e., kernels of orderr for estimation of derivatives of orderm. Convergence theorems are easily derived from interpretation by means of projections inL ² spaces.     

Random vectors satisfying Khinchine–Kahane type inequalities for linear and quadratic forms

We study the behaviour of moments of order p (1 < p < ∞) of affine and quadratic forms with respect to non log‐concave measures and we obtain an extension of Khinchine–Kahane inequality for new families of random vectors by using Pisier's inequalities for martingales. As a consequence, we get some estimates for the moments of affine and quadratic forms with respect to a tail volume of the unit ball of lⁿ_q (0 < q < 1). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enneper immersions

We describe a method to construct minimal surfaces inR ³ which has many computational simplicities and prove that any immersed minimal surface inR ³ may be constructed by using that method. We also show that under certain finite hypotheses on the orders of the coordinate functions, among all conformal minimal immersions of the plane with Gauss mapg=e ^az+b only the parametrizations of helicoids and planes are proper embeddings.     

Weak convergence of linear forms in D[0, 1]

Convergence in probability of the linear forms Σ_k=1^∞a_nkX_k is obtained in the space D[0, 1], where (X_k) are random elements in D[0, 1] and (a_nk) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements (X_n) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D[0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.     

New Expressions for Repeated Upper Tail Integrals of the Normal Distribution

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E(x + i N) ⁿ and E1 / (x ² + N ²) ⁿ , where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated.     

Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ^♯:=[φ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ^♯ are constructed in such a way that φ^♯ has p.p.o.=n>m and their two-scale symbols P^♯ and Q^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ_m^D, then φ^♯:=[φ_m^D,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ^♯=[φ^⊤,φ₃]^⊤ and multi-wavelet ψ^♯ that not only increase the order of vanishing moments but also preserve symmetry. Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday Mathematics subject classifications (2000) 42C15, 42C40. Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512. Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.     

Transfer theorems and asymptotic distributional results for m‐ary search trees

We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m‐ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so‐called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (t_n) [roughly, t_n = O(n^1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, t_n = ω(n^1/2) but t_n = o(n)] we have convergence to nonnormal distributions if m ≤ m₀ (where m₀ ≥ 26) and typically periodic behavior if m ≥ m₀ + 1; and (iii) for large toll sequences [roughly, t_n = ω(n)] we have convergence to nonnormal distributions for all values of m. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

On L 1 Bounds for Asymptotic Normality of Some Weakly Dependent Random Variables

In the present paper, we consider L ₁ bounds for asymptotic normality for the sequence of r.v.’s X ₁,X ₂,… (not necessarily stationary) satisfying the ψ-mixing condition. The L ₁ bounds have been obtained in terms of Lyapunov fractions which, in a particular case, under finiteness of the third moments of summands and the finiteness of ∑ _r≥1 r ² ψ(r), are of order O(n ^−1/2), where the function ψ participates in the definition of the ψ-mixing condition.