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.     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Asymptotic Distribution of Extremes of Randomly Indexed Random Variables

Let {X _n} be a sequence of i.i.d. random variables and let {_k} be a sequence of random indexes. We study the problem of the existence of non-degenerated asymptotic distribution for min{X ₁,..., X _n}.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Extremal problems in classes of univalent functions,omitting prescribed values

Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class R of functions f()=c₁+c₂²+... regular and univalent in the circleU={||<1} and such that f(₁)f(₂)=1 for ₁₂U (the class of Bieberbach-Eilenberg functions). Here one solves the problem of the maximum of |f(₀)| in the class of functions f()R with a fixed value f(₀, where ₀ is an arbitrary point U, and of the maximum of |f(₀)| in the entire class R. For the proof one makes use of the method of the moduli of families of curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 94–114, 1985.     

On ortbogonal polynomial bases in the spacesC andL

, >0 C L - ( ) {Q _n(x)} , Q _n (x)–v _n n ¹⁺ nn ₀ (). , =0.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H₀(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

On finite and totally finite elements in vector lattices

, X , , , M l- M X , ( M) M .