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.

     

Some Asymptotic Results for the M/M/∞ Queue with Ranked Servers

We consider an M/M/ model with m primary servers and infinitely many secondary ones. An arriving customer takes a primary server, if one is available. We derive integral representations for the joint steady state distribution of the number of occupied primary and secondary servers. Letting =/ be the ratio of arrival and service rates (all servers work at rate ), we study the joint distribution asymptotically for . We consider both m=O(1) and m scaled to be of the same order as . We also give results for the marginal distribution of the number of secondary servers that are occupied.     

Asymptotic behavior of constrained stochastic approximations via the theory of large deviations

Summary Let G be a bounded convex set, and _G the projection onto G, and a bounded random process. Projected algorithms of the types , where 0<a _n0, a _n =) occur frequently in applications (among other places) in control and communications theory. The asymptotic convergence properties of {X _n } as 0, n, have been well analyzed in the literature. Here, we use large deviations methods to get a more thorough understanding of the global behavior. Let be a stable point of the algorithm in the sense that X _n in distribution as 0, n. For the unconstrained case, rate of convergence results involve showing asymptotic normality of , and use linearizations about . In the constrained case is often on G, and such methods are inapplicable. But the large deviations method yields an alternative which is often more useful in the applications. The action functionals are derived and their properties (lower semicontinuity, etc.) are obtained. The statistics (mean value, etc.) of the escape times from a neighborhood of are obtained, and the global behavior on the infinite interval is described.Research has been supported in part by the US Army Research Office under Contract #DAAG 29-84-K-0082, and in part by the Office of Naval Research under Contract #N00014-83-K0542Research has been supported in part by the National Science Foundation Grant #ECS 82-11476, and the Air Force Office of Scientific Research under Contract #AF-AFOSR 81-0116     

On shifts of sequences

X ₂ ={x _k } _k=2 X={x _k } _k=1 . , , ( , ) . , , X— , , X ₂ H, , , , , X ₂ H.     

Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP

A large-step infeasible path-following method is proposed for solving general linear complementarity problems with sufficient matrices. If the problem has a solution, the algorithm is superlinearly convergent from any positive starting points, even for degenerate problems. The algorithm generates points in a large neighborhood of the central path. Each iteration requires only one matrix factorization and at most three (asymptotically only two) backsolves. It has been recently proved that any sufficient matrix is a P _*()-matrix for some 0. The computational complexity of the algorithm depends on as well as on a feasibility measure of the starting point. If the starting point is feasible or close to being feasible, then the iteration complexity is . Otherwise, for arbitrary positive and large enough starting points, the iteration complexity is O((1 + )² nL). We note that, while computational complexity depends on , the algorithm itself does not.     

Definition einer Dixmier-Abbildung fürs l(n,C)

Summary Let g be a complex Lie-algebra,G its adjoint algebraic group, and the space of primitive ideals in the enveloping algebra of . For solvable , J. Dixmier has defined a map from the dual space g* into, and R. Rentschler has proved the injectivity of the corresponding map g*/G defined on the space of orbits ofG in g*. The aim of the present paper is to define a similar map g*/G forg=sl(n,C). In a joint paper with J. C. Jantzen we shall prove that it is also injective.     

Spectral approximation of the periodic-nonperiodic Navier-Stokes equations

Summary In order to approximate the Navier-Stokes equations with periodic boundary conditions in two directions and a no-slip boundary condition in the third direction by spectral methods, we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure. The compatibility between these two spaces is checked via an infsup condition. We analyze a spectral and a collocation pseudo-spectral method for the Stokes problem and a collocation pseudo-spectral method for the Navier-Stokes equations. We derive error bounds of spectral type, i.e. which behave likeM ^– whereM depends on the number of degrees of freedom of the method and represents the regularity of the data.     

Exponential inequalities for dependent random variables

This paper contains the Kolmogorov-Prokhorov exponential inequalities for dependent random variables, i.e., for-mixing,-mixing and-mixing. As an application, the law of iterated logarithm is established for stationary-mixing sequence under a nearly best assumption.Research supported by National Science Foundation Grant.     

Approximation of a nondifferentiable nonlinear problem related to MHD equilibria

Summary We present a simple method, based on a variant of the implicit function theorem, which leads to the existence of (a part of) a nontrivial solution branch of the nonlinear eigenvalue problem –u=u ⁺ in ,u=–1 on , where is a two-dimensional domain with boundary . The advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method; the error analysis of the discrete problem appears immediately. We give also an iteration scheme which allows to solve the approximate problem.     

Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations

Summary We discuss the application of a class of spline collocation methods to first-order Volterra integro-differential equations (VIDEs) which contain a weakly singular kernel (t–s)^– with 0<<1. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes. The grading exponents of graded meshes used are not greater thanm (the polynomial degree) which is independent of . This is in contrast to the theories of spline collocation methods for Volterra (or Fredholm) integral equation of the second kind. Numerical examples are given to illustrate the theoretical results.     

Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case

Summary The structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems. The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth ²-level. However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor (where-1/ is the magnitude of the stiff eigenvalues). In these cases a full asymptotic error expansion exists in thestrongly stiff case ( sufficiently small compared to the stepsizeh). The general case (where the oscillating error components areO(h ²) and notO(h ²)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers.     

The theory of an oscillating cylinder viscometer

The theoretical details of periodic state of motion of a liquid (Newtonian) in a narrow annular gap between long vertical coaxial cylinders is given in a new way.If the inner cylinder is suspended by a delicate torsion wire while the outer cylinder undergoes forced harmonic oscillations about its axis, the amplitude of uniform oscillation attained by the inner cylinder can be calculated. Two equations are derived for this purpose; the first is neither simple nor direct with respect to (coefficient of viscosity of the liquid) and the other constants of the apparatus, but covers all liquids with various densities and viscosities, while the second is more simple and direct but covers only liquids of >1. The theory agrees well with experiment for all values of ₀/₀, where ₀ is the amplitude of deflection of the inner cylinder and ₀ is the amplitude of oscillations inexorably imposed on the outer cylinder. ₀/₀ andV _–d/0, whereV _–d is the velocity of the liquid in the neighbourhood of the curved surface of the inner cylinder, has been plotted as function of the frequency for different values of the constants of the liquid of an available apparatus. Also the effect of density of the liquid has been discussed, and the condition of resonance between the liquid and the inner cylinder is given.

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Zusammenfassung Die Arbeit gibt eine theoretische Behandlung der periodischen Bewegung einer Flüssigkeit in der Lücke zwischen zwei koaxialen Zylindern wieder. Ist der innere Zylinder an einem Torsionsdraht aufgehängt, wobei der äussere Zylinder eine erzwungene Schwingung ausführt, wird eine Berechnung der uniformen Schwingung des inneren Zylinders möglich. Die experimentellen Daten sind gut mit der Theorie vereinbar. Die Kurven, welche die Abhängigkeit von ₀/₀ undV _–d/₀ von der Schwingungszahl darstellen, sind für verschiedene Werte der Konstanten der Flüssigkeit und des Apparates gezeichnet, wobei ₀ die Schwingungsamplitude des inneren Zylinders, ₀ diejenige des äusseren Zylinders undV _–d die Geschwindigkeit der Flüssigkeit in der Nähe der krummen Oberfläche des inneren Zylinders bedeutet. Im weiteren wird der Einfluss der Dichte diskutiert; schliesslich werden die Resonanzbedingungen zwischen der Flüssigkeit und dem inneren Zylinder aufgestellt.

     

Collocation for initial value problems

A collocation method for initial value problems, parametrized byn + 1, the number of collocation points, and , the step size, is shown (using Kantorovich's methods) to produce errors which are uniformlyO[/n] ^p+1 for linear time varying systems of ordinary differential equations whose solutions arepth order continuous. Using Wright's method, the single step error is shown to yield errors which areO[ ^n+k+2 for anyk, 0 <k <n, by suitable choice of the collocation points.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation i _t ++(1–²)=0, where is a complex-valued function defined on ^N×, and study the following 2-parameters family of solitary waves: (x, t)=e ^it v(x ₁–ct, x), where and x denotes the vector of the last N–1 variables in ^N . We prove that every distribution solution , of the considered form, satisfies the following universal (and sharp) L -bound:

Temporal Difference Methods for the Maximal Solution of Discrete-Time Coupled Algebraic Riccati Equations

In this paper, we present an iterative technique for deriving the maximal solution of a set of discrete-time coupled algebraic Riccati equations, based on temporal difference methods, which are related to the optimal control of Markovian jump linear systems and have been studied extensively over the last few years. We trace a parallel with the theory of temporal difference algorithms for Markovian decision processes to develop a -policy iteration like algorithm for the maximal solution of these equations. For the special cases in which =0 and =1, we have the situation in which the algorithm reduces to the iterations of the Riccati difference equations (value iteration) and quasilinearization method (policy iteration), respectively. The advantage of the proposed method is that an appropriate choice of between 0 and 1 can speed up the convergence of the policy evaluation step of the policy iteration method by using value iteration.     

The dependence of critical parameter bounds on the monotonicity of a Newton sequence

Summary A new method is proposed for the inclusion of the critical parameter ^* of some convex operator equationu=Tu (appearing e.g. in thermal explosion theory). It is based on the fact that for a fixed Newton's method starting with a suitable subsolution is not monotonically if and only if >^*. Several numerical examples arising from nonlinear boundary value problems illustrate the efficiency of the method.     

Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen

Summary For solving the nonlinear systemG(x, t)=0,G| ⁿ × ^{1 n} , which is assumed to have a smooth curve of solutions a continuation method with self-choosing stepsize is proposed. It is based on a PC-principle using an Euler-Cauchy-predictor and Newton's iteration as corrector. Under the assumption thatG is sufficiently smooth and the total derivative (₁ G(x, t)₂ G(x, t)) has full rankn along the method is proven to terminate with a solution (x ^N , 1) of the system fort=1. It works succesfully, too, if the Jacobians ₁ G(x, t) become singular at some points of , e.g., if has turning points. The method is especially able to give a point-wise approximation of the curve implicitly defined as solution of the system mentioned above.