Doeblin's big limit theorem

Doeblin regarded his paper(²) as his hardest work. The big limit is that ofP(ⁿ)(x, E) asn tends to infinity, in a measurable non-topologized space. An exposition of part one of this paper was published in Ref. 1. This is the exposition of part two, which contains some reparation as well as clarification.Doeblin's great contributions to probability theory were remembered at a conference 50 Years after Doeblin: Developments in the Theory of Markov Chains, Markov Processes and Sums of Random Variables held at Blaubeuren. Germany November 2–7, 1991. Professor Chung contributed this article to further celebrate this occasion. The editor thanks him.     

Path properties of an infinite system of Wiener processes

Let be a Poisson process on ^d of intensity and letW ₁(t),W ₂ (t),..., be a sequence of independent Wiener processes. LetW _i (t)=X _i +W _i (t) whereX ₁,X ₂,..., are the points of . Consider the processess(t)=#{i:X _i (t)1}. These and related processes are studied.     

The motion of a tagged particle and nonhomogeneous media in R1

We analyze the asymptotic behavior of a tagged particle inside an infinite system of identical elastic point masses. The main objective is to study very nonhomogeneous media—particles which are more and more dispersed far from the origin. We suggest that the limit motion of a tagged particle may serve to classify media in the nonhomogeneous case as well as in the homogeneous case.     

Effect of stoichiometry and diffusion on an epoxy/amine reaction mechanism

The bulk phase kinetics of an epoxy (DGEBA) /amine (DDS) thermoset have been studied using DSC, FTIR, and ¹³C-NMR. In the absence of catalyst, the reaction was found to involve a main exothermic reaction between epoxide and amine hydrogen and a side reaction between tertiary amine formed in the main reaction and epoxide. The main reaction was exothermic while the side reaction had no discernable exotherm. Etherification did not occur to any significant extent. Since only the main reaction is exothermic, DSC was very useful for studying the main reaction kinetics. FTIR was used for determining whether epoxide and amine hydrogen were consumed at different rates as a way of following the side reaction. An IR band previously unused by other investigators was used to monitor the amine hydrogen concentration. NMR confirmed the above mechanism by identifying the formation of a quaternary ammonium ion/alkoxide ion pair as a reaction product of tertiary amine and epoxide. This mechanism has been successfully fit to a rate law valid over the entire extent of reaction. The rate constant for the epoxy/amine addition reaction was found to depend on hydroxide concentration (extent), reaction temperature, and glass transition temperature and included contributions from uncatalyzed and autocatalyzed parts. The side reaction (quaternary ammonium ion formation) formed weak bonds which did not affect the overall system T_g. Both reactions were second order. The rate constants for the main reaction first increase with increasing extent due to autocatalysis by hydroxide before decreasing due to the diffusion limit caused by gelation and vitrification. © 1995 John Wiley & Sons, Inc.     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.     

The hydrodynamic limit of a one-dimensional nearest neighbor gradient system

We study the hydrodynamic behavior of a one-dimensional nearest neighbor gradient system with respect to a positive convex potential . In the hydrodynamic limit the density distribution is shown to evolve according to the nonlinear diffusion equation ,(q)/t= (²/dq²){F([1/₁(q)]), with F= –.     

A validated solid-liquid extraction method for the HPLC determination of polyphenols in apple tissues Comparison with pressurised liquid extraction

A solid-liquid extraction procedure followed by reversed-phase high-performance liquid chromatography (RP-HPLC) coupled with a photodiode array detector (DAD) for the determination of polyphenols in freeze-dried apple peel and pulp is reported. The extraction step consists in sonicating 0.5 g of freeze-dried apple tissue with 30 mL of methanol-water-acetic acid (30:69:1, v/v/v) containing 2 g of ascorbic acid/L, for 10 min in an ultrasonic bath. The whole method was validated, concluding that it is a robust method that presents high extraction efficiencies (peel: >91%, pulp: >95%) and appropriate precisions (within day: R.S.D. (n = 5) <5%, and between days: R.S.D. (n = 5) <7%) at the different concentration levels of polyphenols that can be found in apple samples. The method was compared with one previously published, consisting in a pressurized liquid extraction (PLE) followed by RP-HPLC-DAD determination. The advantages and disadvantages of both methods are discussed.     

The laplacian in regions with many small obstacles: Fluctuations around the limit operator

We consider the Laplacian _m in ³ (or in a bounded region of ³) with Dirichlet boundary conditions on the surfaces of some identical (small) neighborhoods ofm randomly distributed points, in the limit whenm goes to infinity and their linear size decreases as 1/m. We give here a stronger form of the result showing the convergence of the above operator to – C(x), whereC(x) is the limit density of electrostatic capacity of the obstacles. In particular results on the rate of convergence and on the fluctuations of _m around the limit operator are given.     

Stability and Asymptotic Analysis of a Fluid-Particle Interaction Model

We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov–Fokker–Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models.     

An empirical process interpretation of a model of species survival

We study a model of species survival recently proposed by Michael and Volkov. We interpret it as a variant of empirical processes, in which the sample size is random and when decreasing, samples of smallest numerical values are removed. Micheal and Volkov proved that the empirical distributions converge to the sample distribution conditioned not to be below a certain threshold. We prove a functional central limit theorem for the fluctuations. There exists a threshold above which the limit process is Gaussian with variance bounded below by a positive constant, while at the threshold it is half-Gaussian.