Large deviation theorem for Hill's estimator

To estimate the exponent of a regularly varying d.f. F, the asymptotic behaviour of Hill's estimator has been extensively discussed. Under the assumption that the d.f. F is continuous, we obtain the large deviation theorem for Hill's estimator. Project supported by the National Natural Science Foundation of China     

A large deviation theorem for weighted sums

Asymptotic representations are derived for large deviation probabilities of weighted sums of independent, identically distributed random variables. The main theorem generalizes a 1952 theorem of Chernoff which asserts that n ^–1 log P(S _n>cn)–log , where S _n is the partial sum of a sequence of independent, identically distributed random variables X ₁, X ₂, ... and is a constant depending on X ₁. The main result is similar in form to, but different in focus from, a particular case of Feller's (1969) theorem on large deviations for triangular arrays.This paper is based on work done for the author's doctoral dissertation written under Prof. Donald R. Truax of the University of Oregon, Eugene.     

The Atkinson-Wilcox expansion theorem for electromagnetic chiral waves

Consider the problem of scattering of a time-harmonic electromagnetic wave by a three-dimensional bounded and smooth obstacle. The infinite space outside the obstacle is filled by a homogeneous isotropic chiral medium. In the region exterior to a sphere that includes the scatterer, any solution of the generalized Helmholtz's equation that satisfies the Silver-Müller radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. The coefficients of the expansion can be determined from the leading coefficient, “the radiation pattern”, by a recurrence relation.     

A strong large deviation theorem

We prove a strong large deviation theorem for an arbitrary sequence of random variables, that is, we establish a full asymptotic expansion of large deviation type for the tail probabilities. An Edgeworth expansion is required to derive the result. We illustrate our theorem with two statistical applications: the sample variance and the kernel density estimator.     

A large deviation limit theorem for multivariate distributions

A local limit theorem for large deviations of $\text{o(n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, where n is the sample size, is developed for multivariate statistics which are more general than standardised means, but which depend on n in much the same way. In particular, the cumulants of the statistic are of the same order in $\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ as those of a standardised mean. The theory is derived under conditions which correspond to those in earlier work by Richter on limit theorems for standardised means and by Chambers on the validity of Edgeworth expansions for multivariate statistics.     

关于任意离散随机序列的一个强偏差定理

引用极限对数似然比的概念作为任意随机序列联合分布与其边缘分布"不相似性"的度量,构造几乎处处收敛的上鞅,讨论了任意离散随机序列的强偏差定理.     

Asymptotic expansion in the central limit theorem for quadratic forms

We consider statistics of the form

A note on Laplace's expansion theorem

A short proof of Laplace's expansion theorem is given. The proof is elementary and can be presented at any level of undergraduate studies where determinants are taught. It is derived directly from the definition so that the theorem may be used as a starting point for further investigation of determinants.     

Central limit theorem for Maxwellian molecules and truncation of the wild expansion

We prove an L¹ bound on the error made when the Wild summation for solutions of the Boltzmann equation for a gas of Maxwellian molecules is truncated at the n^th stage. This gives quantitative control over the only constructive method known for solving the Boltzmann equation. As such, it has recently been applied to numerical computation but without control on the approximation made in truncation. We also show that our bound is qualitatively sharp and that it leads to a simple proof of the exponentially fast rate of relaxation to equilibrium for Maxwellian molecules along lines originally suggested by McKean. © 2000 John Wiley & Sons, Inc.     

Generalized Newton-Puiseux expansion and Abhyankar-Moh semigroup theorem

Inventiones mathematicae -     

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.     

An expansion theorem concerning Wilson functions and polynomials

We prove that a relatively general even function f(x) (satisfying a vanishing condition, and also some analyticity and growth conditions) on the real line can be expanded in terms of a certain function series closely related to the Wilson functions introduced by Groenevelt in 2003. The coefficients in the expansion of f will be inner products in a suitable Hilbert space of f and some polynomials closely related to Wilson polynomials (these are well-known hypergeometric orthogonal polynomials).     

An expansion theorem for two‐dimensional elastic waves and its application

We prove an Atkinson–Wilcox‐type expansion for two‐dimensional elastic waves in this paper. The approach developed on the two‐dimensional Helmholtz equation will be applied in the proof. When the elastic fields are involved, the situation becomes much harder due to two wave solutions propagating at different phase velocities. In the last section, we give an application about the reconstruction of an obstacle from the scattering amplitude. Copyright © 2006 John Wiley & Sons, Ltd.     

The paley-wiener theorem for the radon transform

We prove that the support of a complex-valued function f in ?^k is contained in a convex set K if and only if the support of its Radon transform k(s, ω) is, for each ω, contained in s ≦ S_K (ω); here S_K is the support function of the set K. This theorem is used to determine the propagation speeds of hyperbolic differential equations with constant coefficients, to prove the nonexistence of point spectrum for a certain class of partial differential operators, and to give a simple reduction of Lions' convolution theorem to the one-dimensional convolution theorem of Titchmarsh.     

An extension of guo's theorem on domain compression and expansion

A theorem about calculation of topological degree for set-contraotions is presented, and then eigenvectors and nonzero fixed points of strict-set-contractions are inveatigated.     

The quadratic slice theorem and the equiaffine tube theorem for equiaffine Dupin hypersurfaces

The main purpose of this paper is to investigate the quadraticity of slices (i.e., leaves of curvature foliations) of a non-degenerate equiaffine Dupin hypersurface, where an equiaffine Dupin hypersurface is the notion defined as the equiaffine geometrical version of a (not necessarily complete) Dupin hypersurface.