A note on certain fields of finite transcendence type

Let be a finitely generated extension field of ℚ, andα _i,β_j(1⩽i⩽m,1⩽j⩽n) be some complex numbers. Let (k=1,2,3) be fields obtained by adjoining to the numbers {α _i,β_j exp(α_iβ_j)}, {α_i, exp(α_iβ_j)}, and {exp(α_iβ_j)}, respectively. In the present note the relation between the transcendental degree of over and the transcendence type of over ℚ is given. This work was completed in Dpt. Math., Univ. of Southern Mississippi, Hattiesburg, USA.     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Diffusion times and stability exponents for nearly integrable analytic systems

For a positive integer n and R>0, we set . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H _j ) of the completely integrable Hamiltonian on , with unstable orbits for which we can estimate the time of drift in the action space. These functions H _j are analytic on a fixed complex neighborhood V of , and setting the time of drift of these orbits is smaller than (C(1/ɛ _j )^1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

Almost orthogonal submatrices of an orthogonal matrix

Lett≥1 and letn, M be natural numbers,n<M. Leta=(a _i,j ) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ₂-norms of the columns ofA are uniformly bounded. Namely, for allj Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most such that the matrix , whereA _I =(a _i,j ) _j∈I , acts as a (1+ε)-isomorphism from ℓ ₂ ⁿ into . Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140.     

NOTES ON GLAISHER＇S CONGRUENCES

Let p be an odd prime and let n≥1,k≥0 and r be integers,denote by Bk the kth Bernoulli number,It is proved that(i) If r≥1 is odd and suppose 1≥r+4,then ∑j=1^p-1 1/(np+j)^r=-(2n+1)r(r+1)/2(r+2)Bp-r-2p^2(mod p^3).(ii)If r≥2 is even and suppose p≥r+3, then p-1∑j=1 1/(np+j)^r=r/r+1Bv-r-1p(mod P^2).(iii) p-1∑j=1 1/(np+j)p-2=-(2n+1)p(mod P^2).This result generalizes the Glaisher‘s congruence. As a corollary, a generalization of the Wolsten-holme‘s theorem is obtained.     

Asymptotic behaviour of the phase in non-smooth obstacle scattering

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ ⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω _e =ℝ ⁿ . We can prove that if Ω satisfies a certain geometrical condition, then

Asymptotic properties for median cross-validated nearest neighbor median estimate in nonparametric regression

Consider the nonparametric regression model , whereg is an unknown function to be estimated on [0, 1], are the fixed design points in the interval [0, 1] and is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator is taken as the estimator of the unknown functiong(x). Median cross validation (mev) criterion is employed to select the smoothing parameterh. Leth _π ^* be the smoothing parameter chosen by mev criterion. Under mild regularity conditions, the upper and lower bounds ofh _π ^* , the rate of convergence and the weak consistency of the median cross-validated estimate are obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Foundation of Education of China.     

Asymptotic normality and efficiency of a weighted correlogram

For a process X(t)=Σ _j=1 ^M g _j (t)ξ _j (), where g_j(t) are nonrandom given functions, is a stationary vector-valued Gaussian process, Eξ_k(t) = 0, and Eξ_k(0) Eξ_l(τ) = r _kl(τ), we construct an estimate for the functions r _kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of as T → ∞. We consider the problem of the optimal choice of parameters of the estimate depending on observations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998.     

Moments of a statistic caused by random combinations or random matings

Summary Given two sets of sizek, {α ₁...,α _k} and {β ₁...,β _k} there arek! possible combinations of these two , and suppose there is apriori given a number corresponding to the partnership (α ₁,β _j}. The average of the numbers corresponding to is a random variable, and this paper presents the first five moments of the average, and an application in the study of an isolated human population is demonstrated.     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

An application of the almost-positivity of a class of fourth-order pseudodifferential operators

Using the almost-positivity of a class of fourth-order pseudo-differential operators, we prove the inequality where X₀,X_j ∃ ψ _phg ¹ (Ω), with X_j,X complex-valued, are given. On leave: Department of Mathematics, University of Bologna, Piazza di Porta S.Donato, 5 40127, Bologna, Italy. Supported by N.A.T.O.-C.N.R. fellowship.     

Rates in the empirical bayes estimation problem with non-identical components

This paper considers empirical Bayes estimation of the mean θ of the univariate normal densityf ₀ with known variance where the sample sizesm(n) may vary with the component problems but remain bounded by <∞. Let {(θ _n ,X _n =(X _n,1,...,X _{n, m(n)} ))} be a sequence of independent random vectors where theθ _n are unobservable and iidG and, givenθ _n =θ has densityf _θ ^m(n) . The first part of the paper exhibits estimators for the density of and its derivative whose mean-squared errors go to zero with rates and respectively. LetR ^m(n+1)(G) denote the Bayes risk in the squared-error loss estimation ofθ _n+1 usingX _n+1. For given 0<a<1, we exhibitt _n (X₁,...,X _n ;X _n+1) such that . forn>1 under the assumption that the support ofG is in [0, 1]. Under the weaker condition that E[|θ|^2+γ]<∞ for some γ>0, we exhibitt _n ^* (X ₁,...,X _n ;X _n+1) such that forn>1.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

Domination inequality for martingale transforms of a Rademacher sequence

Letf _n = Σ _k=1 ⁿ v _k r _k ,n=1,…, be a martingale transform of a Rademacher sequence (r _n)and let (r _n ^′ ) be an independent copy of (r _n).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true: In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars (a _n)one has where is theK-interpolation norm between ℓ₁ and ℓ₂. We also derive a new exponential inequality for martingale transforms of a Rademacher sequence. This research was supported in part by an NSF grant and an FRPD grant at NCSU.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093