A Nonlinear Extension of the Coale–Lopez Theorem

For a sequence (T _n)_nof nonlinear positive operators on a real Banach space, certain monotonicity conditions are specified under which asymptotic equality holds for the inhomogeneous iterates T _n T _n-1 T ₁ T ₀. As a consequence, weak ergodicity or even path stability holds in the norm-topology. Finally, a nonlinear extension of the Coale–Lopez theorem in demography is shown.     

Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery

Summary We study the problem of estimating an unknown function on the unit interval (or itsk-th derivative), with supremum norm loss, when the function is observed in Gaussian white noise and the unknown function is known only to obey Lipschitz- smoothness, >k0. We discuss an optimization problem associated with the theory ofoptimal recovery. Although optimal recovery is concerned with deterministic noise chosen by a clever opponent, the solution of this problem furnishes the kernel of the minimax linear estimate for Gaussian white noise. Moreover, this minimax linear estimator is asymptotically minimax among all estimates. We sketch also applications to higher dimensions and to indirect measurement (e.g. deconvolution) problems.Dedicated to R.Z. Khas'minskii for his 60th birthday     

Weakly isomorphic transformations that are not isomorphic

Summary A new method for construction of transformations T _i: (X _i, B _i, _i) , i=1,2, that are factors of each other but that are not measuretheoretically isomorphic is provided. This method uses ergodic product cocycles of the form S ⁱ ₁xS ⁱ ₂x...,, where : XZ ₂ is a cocycle, S belongs to the centralizer of T and T is an ergodic translation on a compact, monothetic group X.     

Representations of distributive lattice-ordered semigroups with binary relations

LetR(, , ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(, , ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, , , ) where (A, , ) is a distributive lattice, (A, ) is a semigroup and is additive w.r.t. . We prove thatDLOS is the variety generated byR(, , ¦), and moreover, if (A, , , ) DLOS then it is representable whenever we disregard one of its operations.Presented by Boris M. Schein.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.     

Borel-Weil-Bott theory for classical Lie superalgebras

The paper is devoted to a systematic construction of the elements of Borel-Weil-Bott theory in the supercase. The main result is a presentation of the cohomology of typical irreducible G-sheaves on G/B, where G is the connected component of the identity in a classical complex Lie supergroup and B G an arbitrary Borel subsupergroup. Also presented are some simple known results concerning the cohomology of irreducible G-sheaves on G/P for a parabolic subsupergroup P.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 32, pp. 71–124, 1988.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

Multiple wiener integrals and nonlinear functionals of a nuclear space valued Wiener process

In this work we develop techniques for the study of nonlinear functionals of a -valued Wiener processW _t, where is the dual of a countably Hilbert nuclear space. We construct stochastic integrals and multiple Wiener integrals of operator-valued processes with respect toW _t. The Wiener decomposition of the space of -valued nonlinear functionals ofW _t is established. We also obtain multiple stochastic integral expansions and representations of -valued nonlinear functionals ofW _t as operator-valued stochastic integrals of Itô type.This research was partially supported by CONACYT grants 22537 and PCEXCNA-040651, and Air Force Office of Scientific Research No. F49620 85 C 0144.Presently at CIMAT, A.P. 402 Guanajuato 36000, GTO, México.     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Minimax risk overl p -balls forl p -error

Summary Consider estimating the mean vector from dataN _n (, ² I) withl _q norm loss,q1, when is known to lie in ann-dimensionall _p ball,p(0, ). For largen, the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp. Obvious exceptions aside, the limiting ratio equals 1 only ifp=q=2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. Whenp, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. We also give asymptotic evaluations of the minimax linear risk. Our results are basic to a theory of estimation in Besov spaces using wavelet bases (to appear elsewhere).     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.     

Some fractal sets determined by stable processes

Summary LetY _i be independent stable subordinators in (, ,P) with indices 0< _i <1 andR _i are the ranges ofY _i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR ₁×R ₂, and whenever ₁ + ₂ 1/2, we deduce results for the vector sumR ₁R ₂={x+y:xR ₁,yR ₂}.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

Stochastic flows on a countable set: Convergence in distribution

Consider a sequenceF ₁,F ₂,... of i.i.d. random transformations from a countable setV toV. Such a sequence describes a discrete-time stochastic flow onV, in which the position at timen of a particle that started at sitex isM _n(x), whereM _n =F _n F _n–1 F ₁. We give conditions on the law ofF ₁ for the sequence (M _n) to be tight, and describe the possible limiting law. an example called the block charge model is introduced. The results can be formulated as a statement about the convergence in distribution of products of infinite-dimensional random stochastic matrices. In practical terms, they describe the possible equilibria for random motions of systems of particles on a countable set, without births or deaths, where each site may be occupied by any number of particles, and all particles at a particular site move together.