Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.

     

The polar phase of Li2Ge4O9 at 298, 150 and 90 K

Dilithium tetragermanate is orthorhombic, space group P2₁ca, at 298 K, and is thus in a polar and probably a ferroelectric state. The structure contains two independent Li, four Ge and nine O atoms, all on general positions with site symmetry 1. Three tetrahedrally coordinated Ge positions form crumpled crankshaft‐like chains, forming sheets within the ac plane, and these are interconnected by the fourth, octahedrally coordinated, Ge sites along the b direction. The GeO₄ tetrahedra and GeO₆ octahedra form a three‐dimensional framework containing two different cavities, hosting the two 4+1‐coordinated Li sites. Cooling to 90 K does not alter the space‐group symmetry; the tetrahedral chains behave as a rigid unit and changes occur mainly within the Li coordination spheres.     

In situ electron spin resonance study of styrene grafting of electron irradiated fluoropolymer films for fuel cell membranes

Three different electron beam irradiated fluoropolymers (ETFE, FEP, and PVDF) as well as their grafting reactions with styrene in different diluents were investigated by means of electron spin resonance (ESR). Depending on the atmosphere during irradiation, ESR spectra of peroxy and alkyl radicals were observed. Radical decay as a function of time and temperature was investigated in the presence and absence of solvent. Grafting levels and number of monomer units per chain were calculated for both types of radicals. Irradiation atmosphere, grafting temperature, and added solvent affect the morphology of the fluoropolymers, in particular the crystalline versus amorphous fractions and their swelling. They thus influence the rate at which the initial radicals at the polymer backbone are reached during the grafting process. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3323–3336, 2006     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

Separation of andrographolide and neoandrographolide from the leaves of Andrographis paniculata using high-speed counter-current chromatography

The bioactive diterpenes andrographolide and neoandrographolide from the leaves of Andrographis paniculata NEES (Acanthaceae) were successfully separated by counter-current chromatography. A single 280-min separation yielded 189 mg of 99.9% andrographolide and 9.5 mg of 98.5% neoandrographolide applying water-methanol-ethyl acetate-n-hexane (2.5:2.5:4:1) solvent system. Structure confirmation was done by electrospray MS, one-dimensional NMR experiments, circular dichroism, optical rotation dispersion, and specific optical rotation [alpha]D.     

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

Blackwell's renewal theorem for certain linear submartingales and coupling

The present work surveys some extensions of Blackwell's renewal theorem for a certain class of linear submartingalesS which have been recently obtained by the author. The basic assumption onS is that their conditional increment distribution functions with respect to some filtration are bounded from above and below by integrable distribution functions. Under a further mean stability condition these random walks turn out to be natural candidates for satisfying Blackwell-type renewal theorems. The latter are derived by employing a coupling argument similar to that which has been used in the i.i.d. case by Lindvallet al. A number of applications are also presented.     

Note on irregularities of distribution

Letf be a periodic function on with period 1, piecewise continuously differentiable, satisfying . For an arbitrary sequence = ( _i ) in [0,1) put and . If then _n (f,) >c· logn holds for some positive constantc (depending onf only) and almost alln. In a certain sense the converse is also true: there is a class of functionsf with such that _n (f,) =o (logn).Support has been received from Netherlands Organization for the Advancement of Pure Research (Z. W. O.).     

Preparative separation of flavonoid glycosides in leaves extract of Ampelopsis grossedentata using high-speed counter-current chromatography

Preparative separation of flavonoid glycosides in leaves extract of Ampelopsis grossedentata was conducted using high-speed counter-current chromatograph (HSCCC) with a solvent system composed of n-hexane-ethyl acetate-methanol-water (1:6:1.5:7.5, v/v). In a single operation, 28 mg of 5,7-dihydroxy-3',4'-trihydroxyflavone-3-O-6'-rhamnose and 18 mg of 5,7-dihydroxy-3',4'-dihydroxyflavone-3-O-6'-rhamnose was obtained from 150 mg of the extract. The chemical structure of the two compounds was elucidated by electrospray ionization (EIS) MS and NMR.     

Eine einfache und flexible Synthese von Pyrrolen aus α,β-ungesättigten Sulfonen

A Simple and Flexible Synthesis of Pyrroles from α , β -Unsaturated Sulfones: The addition of alkyl isocyanoacetates and isocyanoacetonitrile to α,β-unsaturated sulfones affords a convenient and broad access to pyrroles with unusual substitution patterns (see Scheme 2). The α,β-unsaturated sulfones required as starting materials are easily obtained from different olefines.