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.     

Nonparametric Estimation of a Conditional Quantile for α-Mixing Processes

Let (X_i,Y _i) be a set of observations form a stationary -mixing process and (x) be the conditional -th quantile of Y given X = x. Several authors considered nonparametric estimation of (x) in the i.i.d. setting. Assuming the smoothness of FF(x), we estimate it by local polynomial fitting and prove the asymptotic normality and the uniform convergence.     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

The classification of finite connected hypermetric spaces

A finite distance spaceX, d d: X ² is hypermetric (of negative type) if a _x a _y d(x, y) 0 for all integral sequences{a _x x X} that sum to 1 (sum to 0).X, d is connected if the set {(x, y)d(x, y) = 1, x, y X} is the edge set for a connected graph onX, and graphical ifd is the path length distance for this graph. Then we proveThe first author was partially supported by NSF grant DMS 8600882.     

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

Pluriadjoint bundles of polarized surfaces

LetX be a complex connected projective smooth algebraic surface and letL be an ample line bundle onX. The maps associated with the pluriadjoint bundles (K _X L) ¹,t2, are studied by combining an ampleness result forK _X L with a very recent result by Reider. It turns out that apart from some exceptions and up to reductions, 1) (K _X L)³ is very ample; 2) (K _X L) ² is ample and spanned by global sections and is very ample unless eitherg (L)=2 (arithmetic genus ofL) orX contains an elliptic curveE withE ²=0,E·L=1;3) when (K _X L) ² is not very ample, the associated map has degree 4, equality implying thatg (L)=2 and .     

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

The following theorem is going to be proved. Letp _m be them-th prime and putd _m :=p _m+1–p _m . LetN(,T), 1/21,T3. denote the number of zeros =+i of the Riemann zeta function which fulfill and ||T. Letc2 andh0 be constants such thatN(,T)T ^c(1–) (logT) ^h holds true uniformly in 1/21. Let >0 be given. Then there is some constantK>0 such that      

Balanced sets andQ-polynomial association schemes

The notion of a balanced set of vectors is defined, and the classification of such sets suggested. A slightly stronger condition is considered for association schemes, with the following result. LetX be ad-class symmetric association scheme with Bose-Mesner algebraM and Krein parametersq _ij ^h , and let E = E_t (0 t d) be any primitive idempotent ofM. For eachx X letx _E denote the diagonal matrix withy, y entryE _xy(y X). Define the representation diagramD _E on the nodes 0, 1, ...,d by drawing an undirected arc between any nodesh, j for whichq _tj ^h > 0. CallD _E an augmented tree if it has a single loop, whose removal yields a tree. Writexy = i ifx, y X are ith associates (0 i d).     

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.     

Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System (≤ λ)/G/m

We consider a queuing system ()/G/m, where the symbol () means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed dt. The case where the queue length first attains the level r m+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity ₁(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, ₁(t) = O( ^{r+ 1}₁ ^{m– 1} _{r– m+ 1}), where _k is the kth moment of the service-time distribution.     

Estimation of distribution density

The article investigates the asymptotic properties of the estimators of the density P(x) derived using a sample X₁,...,X _n when the distance of the estimator from p(x) is measured in the metricL _p,2p2.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 98, pp. 61–85, 1980.     

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      

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

The Berry-Esseén theorem for strongly mixing Harris recurrent Markov chains

Summary Let ₀, ₁,... be a stationary Harris recurrent Markov chain with state space (E,), and let f E IR, X _i=f(_i). It is known that the sequence X _i, i0, is strongly mixing, i.e. (n)>0 where (n) are the strong (or Rosenblatt) mixing coefficients. If (n) decreases at a sufficiently fast rate and f is suitable, then a central limit holds for the partial sums . The present paper gives conditions in order that the convergence rates are O(n ^–1/2).     

Semilinear Perturbations of Harmonic Spaces,Liouville Property and a Boundary Value Problem

Let (X,) be a P-harmonic Bauer space and let be a Borel measurable function on X×R satisfying conditions (A) through (D) of Section 2 (e.g., (x,t)=t|t|^–1 where >1). For every Kato family M of potential kernels on X let ^M U(X) denote the set of all real continuous functions on X such that u+K ^M _D (,u)(D) for every open relatively compact subset D of X. We study the existence of a non-trivial function in ^M U(X) which is dominated by a given positive harmonic function on X. If X is a domain of R ^d , is a positive Kato measure on X and L is a second-order differential operator in R ^d , we apply our study to derive a characterization of finite positive measures on the minimal Martin boundary ^M ₁ X for which the boundary value problem Lu=(,u) in X and u= on ^M ₁ X is solvable.     

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.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

A combinatorial result about points and balls in euclidean space

For eachn1 there isc _n >0 such that for any finite sexX there isA X, |A|1/2(n+3), having the following property: ifB A is ann-ball, then |B X|c _n |X|. This generalizes a theorem of Neumann-Lara and Urrutia which states thatc ₂1/60.     

On the limit theorems for random variables with values in the spaces L p (2≦p<∞)

Summary We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IEX²= but yet satisfying the Law of the Iterated Logarithm (in short the LIL). We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l _p or L _p with 2<p< which satisfy the CLT. As an application we show that in l _p (2<p<) the stochastic boundedness of the weighed partial sums does not imply the CLT.Research partially supported by NSF Grant MCS 75-07605 A01