Products of independent,normally attracted random variables

Summary LetR _n denote the correlation coefficient of ann-sample of pairs (X _i ,Y _i ), each distributed as (X, Y). AssumeX andY are independent and in the domain of attraction of the Normal law. It is shown that this entailsXY being in that domain of attraction, and thatd _n R _n N(0, 1) in distribution for constantsd _n satisfying lim sup(n ^1/2/d _n )1. Examples illustrate details of these limit theorems.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Remarks on a theorem of Burkholder and Gundy

Let (Y _n) be a sequence of i.i.d. random variables with zero mean such thatP(Y ₁0)>0. Consider the random walkS _n=Y₁+...+Y_n and an a.s. finite stopping time with respect to the -fieldsF _n=(Y₁,..., Y_n). In this paper we present a number of remarks on a theorem of Burkholder and Gundy concerning an estimate involvingE(sup _n1|Svn|^p).Dedicated to the memory of Professor József Mogyoródi.     

On stochastic orders of absolute value of order statistics in symmetric distributions

Let Y₁,…,Y_n be the order statistics of a simple random sample from a finite or infinite population, having median =M. We compare the variables |Y_j−M| and |Y_m−M|, where Y_m is the sample median, that is, for odd n. The comparison is in terms of the likelihood ratio order, which implies stochastic order as well as other orders. The results were motivated by the study of best invariant and minimax estimators for the k/N quantile of a finite population of size N, with a natural loss function of the type , where F_N is the population distribution function, t is an estimate, and g is an increasing function.     

A generalization of the Iterated Logarithm Law for weighted sums with infinite variance

Summary A generalization of the classical Law of the Iterated Logarithm (LIL) is obtained for the weighted i.i.d. case consisting of sequences { _n Y _n } where the weights { _n } are nonzero constants and {Y _n} are i.i.d. random variables. If Y is symmetric but not necessarily square integrable and if the weights satisfy a certain growth rate, conditions are given which guarantee that { _n Y _n} obey a Generalized Law of the Iterated Logarithm (GLIL) in the sense that almost certainly for some positive conslants a _n . Teicher has shown that such weights entail the classical LIL when EY ²< and Feller has treated the GLIL when _n =1 and EY ²=. The main finding here asserts that if {q_n} satisfies q _n ² =nG(q_n)loglogq _n where G is a specified slowly varying function, asymptotically equivalent to the truncated second moment of Y, and if a certain series converges, then the GLIL obtains with where .     

On the number of halving planes

LetS ³ be ann-set in general position. A plane containing three of the points is called a halving plane if it dissectsS into two parts of equal cardinality. It is proved that the number of halving planes is at mostO(n ^2.998).As a main tool, for every setY ofn points in the plane a setN of sizeO(n ⁴) is constructed such that the points ofN are distributed almost evenly in the triangles determined byY.Research supported partly by the Hungarian National Foundation for Scientific Research grant No. 1812     

High density limit theorems for nonlinear chemical reactions with diffusion

Summary The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX _v,N (law of large numbers (LLN)).X _v,N is constructed on a gridS _N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX _v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: , asN , and for the CLT: , asN . The limitY =Y ^X in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions ofY ^X in dependence ofX is also investigated.On leave from Universität Bremen. This work was supported by the Stiftung Volkswagenwerk and a grant from ONR     

Almost Sure Convergence of the Minimum Bipartite Matching Functional in Euclidean Space

Let L _N = L _MBM (X ₁, . . .,X _N ;Y ₁, . . . , Y _N ) be the minimum length of a bipartite matching between two sets of points in R ^d , where X ₁,...,X _N , . . . and Y ₁, . . . , Y _N , . . . are random points independently and uniformly distributed in [0, 1] ^d . We prove that for d 3, L _N /N ^1–1/d converges with probability one to a constant _MBM (d) > 0 as N .     

On Acyclic Orientations and Sequential Dynamical Systems

We study a class of discrete dynamical systems that consists of the following data: (a) a finite (labeled) graph Y with vertex set {1, …, n}, where each vertex has a binary state, (b) a vertex labeled multi-set of functions (F_i, Y: ₂ⁿ →  ₂ⁿ)_i, and (c) a permutation π  S_n. The function F_i, Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F_i, Y are applied. By composing the functions F_i, Y in the order given by π we obtain the sequential dynamical system (SDS):

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In this paper we first establish a sharp, combinatorial upper bound on the number of non-equivalent SDSs for fixed graph Y and multi-set of functions (F_i, Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs.     

Rare numbers

Suppose thatX ₁,X ₂,... is a sequence of iid random variables taking values inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofn ^–(1–) Y _n for [0, 1].     

Discrépances de suites homothétiques

We study the normal order of the discrepancy of the family of all sequencest U, for a given sequenceU of real numbers.U is considered as a point of the probability spaceY of all sequencesU=(u _n ) _n1 such that _n u _n m _n for alln, ( _n ) _n1 and (m _n)_n1 being two fixed sequences of real numbers. Probability onY is the infinite product of uniform probabilities on [ _n ,m _n ]. Assume convergence of the series . Then, with some technical condition, we have for almost all sequenceU inY: whereD _N ^* (V) is the discrepancy of the sequenceV.     

Discrimination with respect to a Gaussian process

Summary Let (N _t) and (Y _t), t in [0,1], be stochastic processes on (, , P). Suppose that (N _t) is Gaussian, m.s. continuous, zero mean, and vanishes a.s. at t=0. Let v _Y and v _N be the induced measures on ^[0,1]. Conditions are obtained for v _Y to be absolutely continuous w.r.t. v _N. Expressions for the Radon-Nikodym derivative are derived. Further results on these problems are obtained for measures induced on L ₂[0, 1] and on C[0, 1].Research supported by ONR Contracts N 00014-75-C-0491 and N 00014-81-K-0373     

Divergence rates for the number of rare numbers

Suppose thatX ₁,X ₂, ... is a sequence of i.i.d. random variables taking value inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY _n /E[Y _n ] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y _n ]. These results extend earlier work by the author.     

The capacity of a channel with arbitrarily varying channel probability functions and binary output alphabet

Summary Let X={1,..., a} be the input alphabet and Y={1,2} be the output alphabet. Let X ^t =X and Y ^t =Y for t=1,2,..., X _n = X ^t and Y _n = Y ^t . Let S be any set, C=={w(·¦·¦)s)¦sS} be a set of (a×2) stochastic matrices w(··¦s), and S ^t=S, t=1,..., n. For every s _n =(s ¹,...,s ⁿ ) S ^t define P(·¦·¦s _n)= w(y ^t ¦x ^t ¦s ^t ) for every x _n=x ¹, , x ⁿX _n and every y _n=(y ¹, , y ⁿ)Y _n. Consider the channel C _n ={P(·¦·¦)s _n )¦s _n S _n } with matrices (·¦·¦s), varying arbitrarily from letter to letter. The authors determine the capacity of this channel when a) neither sender nor receiver knows s _n, b) the sender knows s _n, but the receiver does not, and c) the receiver knows s _n, but the sender does not.Research of both authors supported by the U.S. Air Force under Grant AF-AFOSR-68-1472 to Cornell University.     

The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

Let R(X) = Q[x ₁, x ₂, ..., x _n] be the ring of polynomials in the variables X = {x ₁, x ₂, ..., x _n} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a S _n, we let g In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x ₁, x ₂, ..., x _n} and Y = {y ₁, y ₂, ..., y _n}. The diagonal action of S _n on polynomial P(X, Y) is defined as Let R (X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R *(X, Y) denote the quotient of R (X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for R *(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and R *(X, Y) in terms of their respective bases.     

Co-monotone allocations,Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion

For every integrable allocation (X ₁,X ₂, ...,X _n ) of a random endowmentY= _i ^=1/n X _i amongn agents, there is another allocation (X ₁*,X ₂*, ...,X _n *) such that for every 1in,X _i * is a nondecreasing function ofY (or, (X ₁*,X ₂*, ...,X _n *) areco-monotone) andX _i * dominatesX _i by Second Degree Dominance.If (X ₁*,X ₂*, ...,X _n *) is a co-monotone allocation ofY= _i ^=1/n X _i *, then for every 1in, Y is more dispersed thanX _i * in the sense of the Bickel and Lehmann stochastic order.To illustrate the potential use of this concept in economics, consider insurance markets. It follows that unless the uninsured position is Bickel and Lehmann more dispersed than the insured position, the existing contract can be improved so as to raise the expected utility of both parties, regardless of their (concave) utility functions.     

Explicit stationary distributions for compositions of random functions and products of random matrices

If (Y _n) _n ^=1/ is a sequence of i.i.d. random variables onE=(0,+) and iff is positive onE, this paper studies explicit examples of stationary distributions for the Markov chain (W _n) _n=0 defined byW _n=Y _nf(W _n-1). The case wheref is a Moebius function(ax+b)/(cx+d) leads to products of certain random (2,2) matrices and to interesting random continued fractions. These explicit examples are built with a naive idea by considering genral exponential families onE, especially the families of beta distributions of the first and second kind.     

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

In this paper we introduce and study a cohomology theory {H ⁿ (–,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)} _n0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n3, the functor K(–,n) is right adjoint to the functor _n , where _n (X ) is defined as the fundamental groupoid of the n-loop complex ⁿ (X ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with _i (Y)=0 for all in,n+1 and n3; and also we obtain a classification theorem for those spaces: [–,Y]H ⁿ (–, _n (Y)).     

Spherical 7-Designs in 2 n -Dimensional Euclidean Space

We consider a finite subgroup _n of the group O(N) of orthogonal matrices, where N = 2 ⁿ , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 ⁿ -dimensional Euclidean space R ^N . We prove that representations of the group _n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit _n,2 x ^t of any initial point x on the sphere S _{N – 1} is a 7-design in the Euclidean space of dimension 2 ⁿ .