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.

     

Bibasic analystic functions and discrete ‘bibasic’ hypergeometric series

This paper deals with a study of a class of functions called bibasis analytic functions. Using discrete powerz ⁽ⁿ⁾, discrete bibasic hypergeometric functions have been introduced.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Asymptote of the eigenvalues of a completely continuous operator

It is proved that if(x) is the majorant of the s-numbers of a completely continuous operator A (i.e.,'(x)- 0, S_n(A) (n)) and if there are found numbers [0, 1] and r₀ > 0 such that r⁰ (r)/(r) will be monotonic in (r₀,), then for some > 0,((x) will be a majorant of the eigenvalues of A.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 487–492, October, 1973.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

Modified quasilinearization and optimal initial choice of the multipliers part 2—Optimal control problems

This paper considers the problem of extremizing a functionalI which depends on the statex(t), the controlu(t), and the parameter . The state is ann-vector, the control is anm-vector, and the parameter is ap-vector. At the initial point, the state is prescribed. At the final point, the state and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. A modified quasilinearization algorithm is developed; its main property is a descent property in the performance indexR, the cumulative error in the constraints and the optimum conditions.Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of a scaling factor (or stepsize) in the system of variations. The stepsize is determined by a one-dimensional search so as to ensure the decrease in the performance indexR; this can be achieved through a bisection process starting from = 1. Convergence is achieved whenR becomes smaller than some preselected value.In order to start the algorithm, some nominal functionsx(t),u(t), and multipliers (t), must be chosen. In a real problem, the selection ofx(t),u(t), can be made on the basis of physical considerations. Concerning (t) and , no useful guidelines have been available thus far. In this paper, a method for selecting (t) and optimally is presented: the performance indexR is minimized with respect to (t) and . Since the functionalR is quadratically dependent on (t) and , the resulting variational problem is governed by Euler equations and boundary conditions which are linear.Two numerical examples are presented, and it is shown that, if the initial multipliers (t) and are chosen optimally, modified quasilinearization converges rapidly to the solution. On the other hand, if the initial multipliers are chosen arbitrarily, modified quasilinearization may or may not converge to the solution. From the examples, it is concluded that the beneficial effects associated with the optimal initial choice of the multipliers (t) and lie primarily in increasing the likelihood of convergence rather than accelerating convergence. However, this optimal choice does not guarantee convergence, since convergence depends on the functional being extremized, the differential constraints, the boundary conditions, and the nominal functionsx(t),u(t), chosen in order to start the algorithm.This research, supported by the National Science Foundation, Grant No. GP-18522, is based on Ref. 1. The authors are indebted to Mr. E. E. Cragg for analytical and computational assistance.     

Recurrence-Transience for Self-similar Additive Processes Associated with Stable Distributions

For any stable distribution on the line, recurrence-transience of the selfsimilar additive process {X _t ,t0} with (X ₁)= is determined. Comparison with the stable Lévy process {Y _t ,t0} with (Y ₁)= is made: if is not strictly stable, then {Y _t } is transient but {X _t } is recurrent except the obviously transient case of monotone sample functions.     

Compact sets in the spaceL p (O,T; B)

Summary A characterization of compact sets in L^p (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space L^p (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {f_n} is bounded in L^q(0,T; B) and in L _loc ¹ (0, T; X) and if {f_n/t} is bounded in L _loc ¹ (0, T; Y) then {f_n} is relatively compact in L^p(0,T; B), p相似文献   

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

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.).     

Factorization of One-Generated Formations

A large class of one-generated factorable foliated formations including composition, bicanonical, and other formations is described. V. A. Vedernikov suggested to study foliated formations with directions such that , where and are also some directions, from a unified point of view. We study one-generated factorable foliated formations with directions from the interval [,], where and are the directions of the bicanonical and composition formations, respectively. In particular, all irreducible factorizations of one-generated composition formations are studied.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

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Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

Two Exterior Algebras for Orthogonal and Symplectic Quantum Groups

Let be one of the N ²-dimensional bicovariant first-order differential calculi on the quantum groups O _q (N) or Sp _q (N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra _s is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars bi-invariant 1-form. Moreover, is central in _u and _u is an inner differential calculus.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Threshold theorems for a stochastic model of an epidemic with natural immunization

A stochastic model of an epidemic is investigated, taking account of the removal of ill members of the population (by death, by recovery with immunization, by isolation) and natural immunization. Limiting distributions are found for the size of the epidemic, the number immunized ₁, and their sum, under the assumption that the original number of susceptible individuals n and the number of ill individuals m , while n 1,n ₀< , where and are the coefficients for the contraction of the disease and of immunization respectively.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 385–392, September, 1970.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Combined free and forced laminar convection in internally finned square ducts

Analysis is presented for the heat transfer performance of square ducts with internal fins from each wall in the case of combined free and forced convection by fully developed laminar flow. Numerical results are obtained for the Nusselt number and the pressure drop parameter for various values of finlengths and heat source parameter. For various values of Rayleigh numbers, the Nusselt number increases with the increase in finlength and decreases with the increase in heat source parameter.

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Zusammenfassung Es wird eine Analyse für den Wärmeaustausch von quadratischen Rohren mit inneren Rippen an jeder Wand im Falle einer Kombination von freier und erzwungener Konvektion bei voll entwickelter laminarer Strömung gegeben. Numerische Resultate für die Nusselt-Zahl und den Druckabfall-Koeffizienten für verschiedene Rippenbreiten und Parameter der Wärmequelle werden erhalten. Für einige Werte der Rayleighzahl wächst die Nusselt-Zahl mit der Rippenbreite und fällt mit wachsendem Parameter der Wärmequelle.

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Nomenclature A cross sectional area of the duct - B _2k Bernoulli numbers - c _p specific heat at constant pressure - D _h hydraulic diameter of finless duct - E _n complex constants (20) - F heat source parameter,Q/c _p - F _n () defined by Equation (14) - G(, , , ) Green's function (15, 16) - g gravitational acceleration - H() Heaviside function - h() defined by Equation (22) - i imaginary unit,i ²=–1 - ImW imaginary part ofW - K(,t) kernel of the integral equation, defined by (25) - k thermal conductivity - L pressure drop parameter, –D _h ² (p/x+ _w )/ - l fin length of each fin, Figure (1) - N _u Nusselt number, Equation (32) - p pressure - Q heat generation rate - R() defined by Equation (26) - R _A Rayleigh number, _w gc _p D _h ⁴ /k - ReW real part ofW - T dimensionless temperature, (t–t _w )/(c _p D _h ² /k) - T _mx dimensionless mixed mean temperature, Equation (33) - t fluid temperature - t ₀ reference temperature atx=0 - u local axial velocity - mean axial velocity - V u/ - W complex function defined by Equation (6) - w suffix denoting wall conditions - W ₀ defined by Equation (9) - W ₁ W–W ₀, Equation (18) - x axial coordinate along the length of the duct - y, z cross-sectional coordinates - constant temperature gradient, t/x - coefficient of thermal expansion of the fluid - fluid density - _n - dynamic viscosity - () Dirac delta function - ² Laplacian operator, ²/y ²/²/z ² - , y/D _h ,z/D _h      

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.