On characterizations of distributions via moments of record values

Summary Let {X _n } _n ⁼¹ be a sequence of i.i.d. random variables having continuous distribution F(x) with E|X| ^l+< for some positive integer l and for some >0. It is shown that for any fixed integer N0 the sequence of moments of record values {E(X _L(n) ) ^l } _n=N characterizes F. Furthermore, this result is applied to the weak convergence of continuous distributions.     

Remarks on properties of probability distributions determined by conditional moments

Summary In this paper we consider some properties of rotation — invariant distributions onR ⁿ , which are determined by a form of conditional moment of order >0. In particular we prove that the Gaussian distribution is determined uniquely by its conditional moments and we investigate the related question of finiteness of exponential moments. The case of general >0 appears to be more difficult to analyze than the case =2, studied previously by other authors.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

An upper bound for SLP using first and total second moments

In 1987, J. Dulá considered the problem of finding an upper bound for the expectation of a so-called simplicial function of a random vector and used for this purpose first and total second moments. Under the same moment conditions we consider some different cases of recourse functions and demonstrate how the related moment problems can be solved by solving nonsmooth (unconstrained) optimization problems and thereafter satisfying simple linear constraint systems.     

On a Lower Bound of L p Norms in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive lower bounds of the L _p norms _np for all p, 1p, in the central limit theorem for -mixing random variables with finite sixth-order moments in a strictly stationary case and finite eighth-order moments in a not necessarily stationary one.     

Polling systems with synchronization constraints

We introduce a new service discipline, called thesynchronized gated discipline, for polling systems. It arises when there are precedence (or synchronization) constraints between the order that jobs in different qucues should be served. These constraints are described as follows: There areN stations which are fathers of (zero or more)synchronized stations (children). Jobs that arrive at synchronized stations have to be processed only after jobs that arrived prior to them at their corresponding father station have been processed. We analyze the performance of the synchronized gated discipline and obtain expressions for the first two moments and the Laplace-Stieltjes transform (LST) of the waiting times in different stations, and expressions for the moments and LST of other quantities of interest, such as cycle duration and generalized station times. We also obtain a pscudo conservation law for the synchronized gated discipline, and determine the optimal network topology that minimizes the weighted sum of the mean waiting times, as defined in the pseudo conservation law. Numerical examples are given for illustrating the dependence of the performance of the synchronized gated discipline on different parameters of the network.Supported by a Grant from the France-Israel Scientific Cooperation (in Computer Science and Engineering) between the French Ministry of Research and Technology and the Israeli Ministry of Science and Technology, Grant No. 3321190.     

Method of moments on semigroups

If ( _j ) is a sequence of measures onR ^k having momentss _n ( _j ) of all ordersnN ₀ ^k and if for eachnN ₀ ^k the sequence (s _{n j} )) _jN converges to somet _n R then some subsequence of ( _j ) converges weakly to a measure with moments of all orders satisfyings _n ()=t _n for allnN^0/k . Thisindeterminate method of moments and the continuity theorems in probability theory suggest a common generalization, dealing with a commutative semigroupS, with involution and a neutral element, and measures on the dual semigroupS ^* ofcharacters on S—hermitian multiplicative complex functions not identically zero. In this setting, a continuity theorem holds for measures on the set of bounded characters,⁽²⁾ and an indeterminate method of moments whenS is finitely generated.⁽²⁾ The latter result is generalized in the present paper to the case of arbitraryS. This leads to a generalization of Haviland's criterion for theK-moment problem, and to a continuity theorem for the so-called perfect semigroups.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

On the discrete mean square of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions at 1

There have been many results obtained so far for the mean square of the (absolute) value of the Dirichlet L-function L(s,) in the critical strip 0<<1, especially on the critical line , but relatively few results were known for discrete mean value of |L(1,)|² till W. Zhang had published papers improving the error term step by step, which have recently been superseded by M. Katsurada and K.Matsumoto in which they succeeded in deriving an asymptotic formula for ₀|L(1,)|². The object of our paper is to point out a structural property contained in the formation of the mean square, to find out the niryana–the true body of the above sum.Dedicated to Professor Jean Louis Nicolás on his sixtieth birthdayin final form: 7 October 2003     

Moments of the Mean of Dubins–Freedman Random Probability Distributions

Explicit formulas are given to recursively generate the moments of the mean M for Dubins–Freedman random distribution functions with arbitrary base measure . Using a standard inversion formula for moments of a distribution on the unit interval, the distribution of M is approximated for several natural choices of . The support of the mean is also considered. It is shown that the support of M is connected whenever is concentrated on the vertical bisector of the unit square S, but may have arbitrarily many gaps otherwise.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().     

On ortbogonal polynomial bases in the spacesC andL

, >0 C L - ( ) {Q _n(x)} , Q _n (x)–v _n n ¹⁺ nn ₀ (). , =0.     

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.     

On the bestL p multipoint local approximation

P (f) — , f L ^p - , k . f 02k–2 P (f) 0.     

Inequalities for moments of tails of random variables,with a queueing application

Summary For a non-negative random variable X and 1 such that EX<, E(X-Y) ⁺ /{E(X-y) ₊}⁺ is monotonic-decreasing in y, and hence no smaller than EX . Inequalities for E(X-Y) ₊ E(, 1, y, z0) are also given. This relation enables an inequality of Kingman for the mean waiting time in a stationary GI/G/1 queue to be sharpened.Work done as Visiting Fellow, Department of Statistics, University of Melbourne     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.