On the limit behavior of increments of sums of independent random variables from domains of attraction of asymmetric stable distributions

The asymptotic behavior of increments of sums of independent identically distributed random variables with incremental length (logn) ^p is considered. The laws describing increments of such length are intermediate between the Csög?-Révész law (for large incremental lengths) and the Erdö-Rényi law (for small incremental lengths). A new result for random variables from the domain of normal attraction of asymmetric stable laws with parameter α ε (1, 2) is obtained.     

Limit theorems for increments of compound renewal processes

We derive universal strong limit theorems for increments of compound renewal processes which unify the strong law of large numbers, Erd?s-Rényi law, Csörg?-Révész law, and law of the iterated logarithm for such processes. New results are obtained under various moment assumptions on distributions of random variables generating the process. In particular, we study the case of distributions from domains of attraction of the normal law and completely asymmetric stable laws with index α ∈ (1, 2). Bibliography: 15 titles.     

Lambert's W, infinite divisibility and Poisson mixtures

The Lambert W function is shown to be the Laplace exponent of a positive infinitely divisible law (i.e. W is a Bernstein function) called the standard Lambert law. This law is a generalized gamma convolution. At least three Poisson mixture families are defined in terms of W. One of these is the generalized Poisson laws which are shown to be generalized negative-binomial convolutions. Mixing with positive stable laws yields further generalizations.     

行星悬浮轨道附近的编队

研究了不同类型的地球和火星悬浮轨道附近的相对运动．首先,推导了悬浮轨道附近的相对运动方程并将相对运动方程在悬浮轨道附近线性化．利用两种半自然编队控制率进行编队控制,其中一种为被动控制,对工程应用有很大的价值．在两种控制率下,讨论了每种悬浮轨道的稳定区域．由于两种控制率都不能满足特殊的火星悬浮轨道任务要求,于是,根据该任务的要求设计了一种特殊的半自然控制率．该控制率既能满足编队任务的要求也能使相对运动稳定．     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.     

New Solution of Famous Theorem

About 1923,the great mathematician Paul Lévy invented a family of probabilitydistributions(laws)called"stable".If X and Y are independent random variableswith the law L,then for any constants a>0 and b,there exist constants c>0 andd such that the law of aX+bY is the same as cZ+d,where Z is a random variable     

Convergence in law of random sums with non-random centering

We extend results obtained in Kruglov,⁽⁷⁾ and Finkelstein and Tucker⁽³⁾ to obtain necessary and sufficient conditions for convergence in law of random sums of non-identically distributed independent random variables under non-random centering. Thei.i.d. case is also considered for random variables attracted to a stable law. Necessary and sufficient conditions for convergence in law of these random variables under non-random centering, and in some cases, under non-random norming, are also obtained. The distribution functions for the limit laws are determined as well, generalizing results of Robbins.⁽¹⁰⁾ Supported in part by The State University of New York and United States Information Agency Grant No. IA AEMP69193692.     

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-K?hler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.     

Tempered stable laws as random walk limits

Stable laws can be tempered by modifying the Lévy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random walk models can provide a basic physical model for the underlying physical phenomena.     

Self-Normalized LIL for Hanson–Russo Type Increments

Let X ₁, X ₂,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given.     

两两NQD列大数定律的一个注记

利用两两NQD列三级数定理的思想和Chebyshev不等式,研究了两两NQD列在一类广泛条件下的弱大数定律和一类加强条件下的强大数定律,得到了与独立情形一致的结果,还特别讨论了同分布情形,推广了相关文献的结果．     

On a Berry-Esseen theorem for compound Poisson processes

The well-known Berry-Esseen theorem concerning the rate of convergence to a stable law for a sum of independent identically distributed (i.i.d.) random variables is adapted to the case of a compound Poisson process, considered in the collective risk theory. As a consequence the rate of convergence of the Edgeworth expansion to the compound Poisson distribution is examined for all positive values of the time variable, in both cases where the moments of the claim distribution converge or diverge. As a by product the results obtained by T. Höglund [1] concerning the sum of a fixed number (n) of i.i.d. random variables are presented in an alternative manner. His theorems concerning the limiting behaviour for n → ∞ can be transformed slightly in order to make them hold for all n. It is explained how the result on the estimation of the rate of convergence in a limit theorem with a stable law fits with the results obtained by K.I. Satyabaldina [2].     

A geometric Schur–Weyl duality for quotients of affine Hecke algebras

After establishing a geometric Schur–Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.     

Size-biased sampling of Poisson point processes and excursions

Summary Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.Research supported in part by NSF grant DMS88-01808 and DMS91-07351     

On quotients of Coxeter groups under the weak order

We consider quotients of finitely generated Coxeter groups under the weak order. Björner and Wachs proved that every such quotient is a meet semi-lattice, and in the finite case is a lattice [Björner and Wachs, Trans. Amer. Math. Soc. 308 (1988) 1–37]. Our result is that the quotient of an affine Weyl group by the corresponding finite Weyl group is a lattice, and that up to isomorphism, these are the only quotients of infinite Coxeter groups that are lattices. In this paper, we restrict our attention to the non-affine case; the affine case appears in [Waugh, Order 16 (1999) 77–87]. We reduce to the hyperbolic case by an argument using induced subgraphs of Coxeter graphs. Within each quotient, we produce a set of elements with no common upper bound, generated by a Maple program. The number of cases is reduced because the sets satisfy the following conjecture: if a set of elements does not have an upper bound in a particular Coxeter group, then it does not have an upper bound in any Coxeter group whose graph can be obtained from the graph of the original group by increasing edge weights.     

A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented.     

A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented. (Received 31 January 2000; in revised form 5 April 2000)     

Strong Limit Theorems for Increments of Sums of Independent Random Variables

We derive universal strong laws for increments of sums of independent, nonidentically distributed, random variables. These results generalize universal results of the author for the i.i.d. case which include the strong law of large numbers, law of the iterated logarithm, Erdos-Renyi law, and Csorgo-Revesz laws. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 260–285.     

NA随机变量列一类强大数律的充分必要条件

本文给出了具有不同分布NA随机变量列满足一类强大数律的充分必要条件, 从而将Egorov对独立随机变量列建立的结果推广到NA随机变量情形; 作为应用, 我们还建立了一个新的强大数律.     

Variation of geometric invariant theory quotients

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.