Precise asymptotics in Spitzer and Baum-Katz's law of large numbers: the semistable case

Let X₁,X₂,… be i.i.d. random variables with distribution μ and with mean zero, whenever the mean exists. Set S_n=X₁+?+X_n. In recent years precise asymptotics as ε↓0 have been proved for sums like ∑_n=1^∞n⁻¹P{|S_n|?εn^1/p}, assuming that μ belongs to the (normal) domain of attraction of a stable law. Our main results generalize these results to distributions μ belonging to the (normal) domain of semistable attraction of a semistable law. Furthermore, a limiting case new even in the stable situation is presented.     

Precise asymptotics in the self-normalized law of the iterated logarithm

Let X,X₁,X₂,… be i.i.d. nondegenerate random variables with zero means, and . We investigate the precise asymptotics in the law of the iterated logarithm for self-normalized sums, S_n/V_n, also for the maximum of self-normalized sums, max_1kn|S_k|/V_n, when X belongs to the domain of attraction of the normal law.     

φ-混合重尾随机向量序列的Chover型重对数律

本文讨论同分布的φ-混合随机向量序列其共同分布属于某个没有Gauss分量的广义的半稳定律的吸引场部分和的积分检验的极限结果,由此可推出相应的Chover型重对数律.     

Invariance principles for generalized domains of semistable attraction

Let X,_X1,_X2,…$X,X_1,X_2,\dots$ be independent and identically distributed R^d${\mathbb{R}}^d$-valued random vectors and assume X$X$ belongs to the generalized domain of attraction of some operator semistable law without normal component. Then without changing its distribution, one can redefine the sequence on a new probability space such that the properly affine normalized partial sums converge in probability and consequently even in L^p$L^p$ (for some p>0$p>0$) to the corresponding operator semistable Lévy motion.     

On the rate of convergence of sums of extremes to a stable law

Summary LetX ₁,X ₂, ..., be a sequence of independent and identically distributed random variables in the domain of normal attraction of a nonnormal stabler law. It is known that only the sum of thek _n largest andk _n smallest extreme values in thenth partial sum withk _n andk _n /n0 are responsible for the asymptotic stable distribution of the whole sum. We investigate the rate at which such sums of extreme values converge to a stable law in conjunction with the rate at which the sums of the middle terms become asymptotically negligible. In terms of rates of convergence our results provide in many cases a quantitative measure of exactly what portion of the sample is asymptotically stable.Research partially supported by the Deutsche Forschungsgemeinschaft while visiting the University of DelawareResearch partially supported by NSF Grant no. DMS-8803209     

On sample path properties of semistable processes

This paper contains three main results: In the first result a correspondence principle between semistable measures on L_p, 1 ≤ p < ∞, and Banach space valued semistable processes is established. In the second result it is shown that the paths of a Banach space valued semistable process belong to L_p with probability zero or one, and necessary and sufficient conditions for the two alternatives to hold are given. In the third result necessary and sufficient conditions are given for almost sure path absolute continuity for certain Banach space valued semistable processes.     

Merging asymptotic expansions for semistable random variables

Merging asymptotic expansions are established for distribution functions from the domain of geometric partial attraction of a semistable law. The length of the expansion depends on the exponent of the semistable law and on the characteristic function of the underlying distribution. We obtain sufficient conditions for the quantile function in order to get real infinite asymptotic expansion. The results are generalizations of the existing theory in the stable case.     

Levi problem and semistable quotients

A complex space X is in class 𝒬 _G if it is a semistable quotient of the complement to an analytic subset of a Stein manifold by a holomorphic action of a reductive complex Lie group G. It is shown that every pseudoconvex unramified domain over X is also in 𝒬 _G .     

A form of multivariate gamma distribution

Let V _i, i=1,..., k, be independent gamma random variables with shape i, scale , and location parameter i, and consider the partial sums Z ₁=V ₁, Z ₂=V ₁+V ₂,..., Z _k=V ₁+...+V _k. When the scale parameters are all equal, each partial sum is again distributed as gamma, and hence the joint distribution of the partial sums may be called a multivariate gamma. This distribution, whose marginals are positively correlated has several interesting properties and has potential applications in stochastic processes and reliability. In this paper we study this distribution as a multivariate extension of the three-parameter gamma and give several properties that relate to ratios and conditional distributions of partial sums. The general density, as well as special cases are considered.     

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andU _L ′, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification ofU′.     

On a functional equation generalizing the class of semistable distributions

With ?(p),p≥0 the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation ?(p-e ^?pβΠ _i=1 ^m ?^γi (c _i p) for somem≥1,c _i>0, γ _i >0,i=1., …,m, β ε ®. We supply its complete solutions in terms of semistable distributions (the ones obtained whenm=1). We then show how to obtain these solutions as limit laws (r → ∞) of normalized Poisson sums of iid samples when the Poisson intensity λ(r) grows geometrically withr.     

A law of the iterated logarithm for semistable laws on vector spaces and homogeneous groups

Let X₁, X₂, ... be a sequence of independent and identically distributed (i.i.d.)R ^d-valued random vectors distributed according to a full (B,c) semistable law without Gaussian component. Then the following law of the iterated logarithm holds.

On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power

Let R _k (n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R ₃ (n) n ^5/9+, and that R ₄ (n) n ^47/90+, where > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.     

Approximation Properties of the Operators ⋎n+2r(f) and of Their Discrete Analogs

This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier--Legendre sums of order n with 2r terms of the form _k=1 ^2r a_kP_n+k(x) added; here P _m(x) denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval [-1,1], which, in fact, for r= = 1 allows us to significantly improve the approximation properties of partial Fourier--Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions and A _q (B). With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties.     

Extreme values and a Gaussian central limit theorem

Summary We examine the central limit theorem with Gaussian limit law for a sequence of independent, identically distributed, vector valued random variables whose partial sums can be centered and normalized to be tight with non-degenerate limit laws. These results apply to the situation when the sequence is in the domain of attraction of a non-degenerate stable law of indexp(0,2], and are achieved by eliminating the extreme values from the partial sums.Supported in part by NSF Grant MCS-8219742Work done while visiting the University of Wisconsin, Madison, with partial support by NSF Grant MCS-8219742     

Partial Sum Process for Records

Suppose the upper records from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z_n(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z_n} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal. AMS 2000 Subject Classification Primary—60F17 Secondary—60G70     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Invariance Principle Under Self-Normalization for Nonidentically Distributed Random Variables

Let V _n ^–1 _n be the adaptive process of self-normalized partial sums S _k of independent random variables X _i , defined by linear interpolation between the points (V _k ²/V _n ²,S _k /V _n ), kn, where V _k ²= _ik X _i ². We prove that if the X _k 's are symmetric, V _n ^–1 _n converges weakly to the Brownian motion W in each Hölder space supporting W if and only if V _n ^–1 max _kn |X _k |=o _P (1). We give some partial extension to the non symmetric case.