Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

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}.     

On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Convergence of epigraphs and of sublevel sets

Let LSC(X) be the set of the proper lower semicontinuous extended real-valued functions defined on a metric spaceX. Given a sequence f _n in LSC(X) and a functionf LSC(X), we show that convergence of f _n tof in several variational convergence modes implies that for each , the sublevel set at height off is the limit, in the same variational sense, of an appropriately chosen sequence of sublevel sets of thef _n, at height _n approaching . The converse holds true whenever a form of stability of the sublevel sets of the limit function is verified. The results are obtained by regarding a hyperspace topology as the weakest topology for which each member of an appropriate family of excess functionals is upper semicontinuous, and each member of an appropriate family of gap functionals is lower semicontinuous. General facts about the representation of hyperspace topologies in this manner are given.     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Spectrum of multidimensional periodic operators

Let be a lattice in the n-dimensional Euclidean space Rⁿ and let F be the fundamental domain of the lattice . We denote by H the Schrödinger operator generated in L₂(Rⁿ) by the expression –u + q(x)u(1), and by H^t the operator generated in L₂(F) by the expression (1) and by quasiperiodic boundary conditions, where q(x) is a periodic (with respect to the lattice ) function. Asymptotic formulas for the eigenvalues of the operator H_t are obtained and with the aid of these formulas it is proved that there exists a number (q) such that the interval [(q), ] belongs to the spectrum of the operator H [for n3 in the case of sufficiently smooth potentials q(x), while for n=2 for any potential q(x) from L₂(F)], i.e., the Bethe-Sommerfeld conjecture is proved for arbitrary lattices.Translated from Teoriya Funktsii, Funktsionali'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 17–34, 1988.     

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.     

A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a _in} are uniformly bounded,E|X|^1/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.     

Semilattices of left reversible semigroups

Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.     

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.     

Monotonic subsequences in permutations of n natural numbers

Let S_n be the set of all permutations of the numbers 1, 2,..., n, and letl _n() be the number of terms in the maximal monotonic subsequence contained in S_n. If M[l _n()] is the mean value ofl _n () on S_n, then, for all except a finite number of n, the bound M[l _n()] e n is valid.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 511–514, April, 1973.The author wishes to thank E. M. Nikishin for having posed the problem and for his constant interest in the work.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

On extremal contractions from threefolds to surfaces: The case of one non-Gorenstein point and a nonsingular base surface

In this paper, we study the local structure of extremal contractionsfXS from threefoldsX with only terminal singularities onto a surfaceS. If the surfaceS is nonsingular andX has a unique non-Gorenstein point on a fiber, we prove that either the linear system ¦–K _X ¦, ¦–2K _X ¦, or ¦–3K _X ¦contains a good divisor.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 45, Algebraic Geometry-8, 1997.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

An improved asymptotic analysis of the expected number of pivot steps required by the simplex algorithm

Leta ₁ ...,a _m be i.i.d. points uniformly on the unit sphere in ⁿ ,m n 3, and letX:= {x ⁿ |a _i ^T x1} be the random polyhedron generated bya ₁, ...,a _m . Furthermore, for linearly independent vectorsu, in ⁿ , letS _u , (X) be the number of shadow vertices ofX inspan(u,). The paper provides an asymptotic expansion of the expectation value¯S _n,m := _in4 ¹ E(S _u, ) for fixedn andm .¯S _n,m equals the expected number of pivot steps that the shadow vertex algorithm — a parametric variant of the simplex algorithm — requires in order to solve linear programming problems of type max u ^T ,xX, if the algorithm will be started with anX-vertex solving the problem max ^T ,x X. Our analysis is closely related to Borgwardt's probabilistic analysis of the simplex algorithm. We obtain a refined asymptotic analysis of the expected number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data.     

Cohomology of Compact Locally Symmetric Spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S()=X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S _H()=Y of X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ⁿ and ^m , then low degree cohomology classes on the variety S() restrict non-trivially to the subvariety S _H (); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S().     

Asymptotically minimax tests of composite hypotheses

Summary X ₁,,X> _n are independent, identically distributed random variables with common density function f(x¦ ₁ ,, _k , _k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that _k+1 =b against the alternative that _k+1 =b+cn ^–1/2 . ₁ ,, _k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦ ₁ -a _i ¦n ^–1/2 M(n) for i=1,,k, where a ₁ , ,a _k are known, and M(n) approaches infinity as n increases but n ^–1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to _k+1 =b, and weight 1-A to _k+1 =b+cn ^–1/2 , in each case the weight being spread uniformly over the possible values of ₁ ,, _k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of ₁ ,..., _k are substituted for a ₁ ..., a _k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a ₁ ..., a _k are.Research supported by the U.S. Air Force under Grant AF-AFOSR-68-1472.     

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.