A De Finetti-type theorem for nonexchangeable finite-valued random variables

We introduce the concept of a random assignment process (roughly speaking, such a process assigns, according to some weight function, labels to the points of an arbitrary set), and we impose conditions on the weight function ensuring that a De Finetti-type theorem is satisfied. In particular, this provides a wide class of finite-valued random variables which, despite they are neither exchangeable nor identically distributed, verify a De Finetti-type theorem (i.e., they are conditionally independent).     

Subadditive ergodic theorems for random sets in infinite dimensions

We prove pointwise and mean versions of the subadditive ergodic theorem for superstationary families of compact, convex random subsets of a real Banach space, extending previously known results that were obtained in finite dimensions or with additional hypotheses on the random sets. We also show how the techniques can be used to obtain the strong law of large numbers for pairwise independent random sets, as well as results in the weak topology.     

Central limit theorems for random polytopes

Let K be a smooth convex set. The convex hull of independent random points in K is a random polytope. Central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity. One essential step is to determine the precise asymptotic order of the occurring variances. Research was supported in part by the European Network PHD, MCRN-511953.     

Necessary and sufficient conditions for quadratic minimality

We provide necessary and sufficient conditions for a (non-convex) quadratic function to take a local minimum over a convex set. Various limiting examples are given.     

Necessary and sufficient optimality conditions for control of piecewise deterministic markov processes

Piecewise deterministic Markov processes (PDPs) are continuous time homogeneous Markov processes whose trajectories are solutions of ordinary differential equations with random jumps between the different integral curves. Both continuous deterministic motion and the random jumps of the processes are controlled in order to minimize the expected value of a performance criterion involving discounted running and boundary costs. Under fairly general assumptions, we will show that there exists an optimal control, that the value function is Lipschitz continuous and that a generalized Bellman-Hamilton-Jacobi (BHJ) equation involving the Clarke generalized gradient is a necessary and sufficient optimality condition for the problem.     

Necessary and sufficient conditions for expansions of Gabor type

In this paper, we give necessary and sufficient conditions for two families of Gabor functions of a certain type to yield a reproducing identity on L^2（R^n）. As applications, we characterize when such families yield orthonormal or bi-orthogonal expansions. We also obtain a generalization of the Balian-Low theorem for general reprodueing identities （not necessary coming from a frame）.     

Approximation theorems for random permanents and associated stochastic processes

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established. The results are based on the martingale decomposition of a random permanent function similar to the one known for U-statistics and on relating the components of this decomposition to some multiple stochastic integrals.Mathematics Subject Classification (2000): Primary 60F17, 62G20; Secondary 15A15, 15A52Acknowledgement A significant part of this work was completed when the first author was visiting the Center for Mathematical Sciences at the University of Wisconsin-Madison. He would like to express his gratitude to the Center and its Acting Director, Prof. Thomas G. Kurtz, for their hospitality. Thanks are also due to the first authors current host, the Institute for Mathematics and Its Applications at the University of Minnesota. Finally, both authors graciously acknowledge the comments of an anonymous referee on an earlier version of this paper.     

Necessary and sufficient conditions for the Robbins-Monro method

This paper examines the relation between convergence of the Robbins-Monro iterates $\text{X}{}_{\mathrm{n+1}}\text{= X}{}_{\mathrm{n}}\text{?a}{}_{\mathrm{n}}\text{?(X}{}_{\mathrm{n}}\text{)+a}{}_{\mathrm{n}}\text{ξ}{}_{\mathrm{n}}\text{, ?(θ)=0}$, and the laws of large numbers S_n=a_nΣ^n?1_j=0ξ_j→0 as n→+∞. If a_n is decreasing at least as rapidly as c/n, then X_n→θw.p. 1 (resp. in L_p, p?1) implies S_n→0 w.p. 1 (resp. in L_p, p?1) as n→+∞. If a_n is decreasing at least as slowly as c?n and lim_n→+∞a_n=0, then S_n→0 w.p. 1 (resp. in L_p, p?2) implies X_n→θw.p. 1 (resp. in L_p, p?2) as n →+∞. Thus, there is equivalence in the frequently examined case $\text{a}{}_{\mathrm{n}}\text{?c?n}$. Counter examples show that the LLN must have the form of S_n, that the rate of decrease conditions are sharp, that the weak LLN is neither necessary nor sufficient for the convergence in probability of X_n to θ when $\text{a}{}_{\mathrm{n}}\text{?c?n}$.     

Limit theorems for sums of random exponentials

We study limiting distributions of exponential sums as t→∞, N→∞, where (X_i) are i.i.d. random variables. Two cases are considered: (A) ess sup X_i = 0 and (B) ess sup X_i = ∞. We assume that the function h(x)= -log P{X_i>x} (case B) or h(x) = -log P {X_i>-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form , where the rate function H₀(t) is a certain asymptotic version of the function (case B) or (case A). We have found two critical points, λ₁<λ₂, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ₂, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.Mathematics Subject Classification (2000): 60G50, 60F05, 60E07     

Local paraconvexity and local selection theorems

Theorems on the local extendability of selections for non-convex-valued maps of paracompact spaces into Banach spaces, i.e., infinite-dimensional analogs of the finite-dimensional Michael selection theorem are proved. We were able to obtain these results under an appropriate metric control of the local degree of nonconvexity on the valuesF(x), which naturally leads us to introduce the notion of equi-locally paraconvex families of sets. It is shown that all convex subsets of the integral curves of the differential equationy′=f(x,y) with a continuous right-hand sidef and the isometric images of such subsets form an equi-locally paraconvex family of subsets of a Euclidean space. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 261–269, February, 1999.     

Necessary and sufficient conditions for a matrix to be causative in a nonstationary Markov chain

Given a sequence of transition matrices for a nonstationary Markov chain, a matrix whose product on the right of a transition matrix yields the next transition matrix is called a causative matrix. A causative matrix is strongly causative if successive products continue to yield stochastic matrices. This paper presents necessary and sufficient conditions for a matrix to be causative and strongly causative with respect to an invertible transition matrix, by considering the causative matrix as a linear transformation on the rows of the transition matrix.     

On alternative theorems and necessary conditions for efficiency

In this article, we establish theorems of the alternative for a system described by inequalities, equalities and a set inclusion, which are generalizations of Tucker's classical theorem of the alternative, and develop Kuhn–Tucker necessary conditions for efficiency to mathematical programs in normed linear spaces involving inequality, equality and set constraints with positive Lagrange multipliers of all the components of objective functions.     

Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems

Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions to the boundary value problem

     

Strong consistency and CLT for the random decrement estimator

The random decrement technique （RDT）, introduced in the sixties by Cole [NASA CR- 2005, 1973], is a very performing method of analysis for vibration signature of a structure under ambient loading. But the real nature of the random decrement signature has been misunderstood until now. Moreover, the various interpretations were made in continuous time setting, while real experimental data are obtained in discrete time. In this paper, the really implemental discrete time algorithms are studied. The asymptotic analysis as the number of triggering points go to infinity is achieved, and a Law of Large Numbers as well as a Central Limit Theorem is proved. Moreover, the limit as the discretization time step goes to zero is computed, giving more tractable formulas to approximate the random decrement. This is a new approach of the famous ＂Kac-Slepian paradox＂ [Ann. Math. Star., 30, 1215-1228 （1959）]. The main point might be that the RDT is a very efficient functional estimator of the correlation function of a stationary ergodic Gaussian process. Very fast, it is to classical estimators what Fast Fourier Transform （FFT） is to ordinary Discrete Fourier Transforms.