Asymptotics for geometric location problems over random samples

1999 ◽  
Vol 31 (03) ◽  
pp. 632-642
Author(s):  
K. McGivney ◽  
J. E. Yukich
Keyword(s):  
Finite Number ◽  
Positive Constant ◽  
Complete Convergence ◽  
Location Problem ◽  
Random Variables ◽  
Location Problems ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Random Samples ◽  
Image Position

Consider the basic location problem in which k locations from among n given points X 1,…,X n are to be chosen so as to minimize the sum M(k; X 1,…,X n ) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the X i , i ≥ 1, are i.i.d. random variables with values in [0,1] d and when k = ⌈n/(D+1)⌉ we show that where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X 1, and c.c. denotes complete convergence.

Asymptotics for geometric location problems over random samples

1999 ◽  
Vol 31 (3) ◽  
pp. 632-642 ◽  
Author(s):  
K. McGivney ◽  
J. E. Yukich
Keyword(s):  
Finite Number ◽  
Positive Constant ◽  
Complete Convergence ◽  
Location Problem ◽  
Random Variables ◽  
Location Problems ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Random Samples ◽  
Image Position

Consider the basic location problem in which k locations from among n given points X1,…,Xn are to be chosen so as to minimize the sum M(k; X1,…,Xn) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the Xi, i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X1, and c.c. denotes complete convergence.

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 337-342 ◽  
Author(s):  
M. S. Srivastava
Keyword(s):  
Random Variables ◽  
Nonparametric Tests ◽  
Absolutely Continuous ◽  
Bivariate Case ◽  
Image Position ◽  
Tests For Independence ◽  
Mutually Independent

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

A d-DEFORMATION OF FREE GAUSSIAN RANDOM VARIABLES

2005 ◽  
Vol 08 (04) ◽  
pp. 669-680 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Tensor Product ◽  
Hypergeometric Series ◽  
Fock Space ◽  
Recurrence Formula ◽  
Random Variables ◽  
Product Formula ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Gaussian Random Variables ◽  
Special Case

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product [Formula: see text]. Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

A local limit theorem for attractions under a stable law

1980 ◽  
Vol 87 (1) ◽  
pp. 179-187 ◽  
Author(s):  
Sujit K. Basu ◽  
Makoto Maejima
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Stable Law ◽  
Absolutely Continuous ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

AbstractLet {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.

scholarly journals Sequential Tests for Normal Markov Sequence

1971 ◽  
Vol 12 (4) ◽  
pp. 433-440 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Markov Chain ◽  
Finite Number ◽  
Transition Probabilities ◽  
Random Variables ◽  
Important Case ◽  
Fundamental Identity ◽  
Markov Sequence ◽  
Sequential Tests ◽  
Image Position

This is a sequel to the author's (Phatarfod [9]) paper in which an analogue of Wald's Fundamental Identity (F.I.) for random variables defined on a Markov chain with a finite number of states was derived. From it the sampling properties of sequential tests of simple hypotheses about the parameters occurring in the transition probabilities were obtained. In this paper we consider the case of continuous Markovian variables. We restrict our attention to the practically important case of a Normal Markov sequence X0,X1,X2,… such that the Yr being independent normal variables with mean zero and variance σ2.

Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

2014 ◽  
Vol 58 (1) ◽  
pp. 125-147 ◽  
Author(s):  
N. A. Chernyavskaya ◽  
L. A. Shuster
Keyword(s):  
Liouville Equation ◽  
Positive Constant ◽  
Absolute Constant ◽  
Absolutely Continuous ◽  
Weighted Estimates ◽  
Negative Function ◽  
Almost Everywhere ◽  
Absolute Positive Constant ◽  
Sturm Liouville ◽  
Image Position

AbstractConsider the equationwhereƒ∈Lp(ℝ),p∈ (1, ∞) andBy a solution of (*), we mean any functionyabsolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the spaceLp(ℝ), i.e.(1) for any function, there exists a unique solutiony∈Lp(ℝ) of (*);(2) there exists an absolute constantc1(p) > 0 such that the solutiony∈Lp(ℝ) of (*) satisfies the inequalityWe study the following problem on the strengthening estimate (**). Let a non-negative functionbe given. We have to find minimal additional restrictions forθunder which the following inequality holds:Here,yis a solution of (*) from the classLp(ℝ), andc2(p) is an absolute positive constant.

ELECTRONIC SPECTRAL AND WAVEFUNCTION PROPERTIES OF ONE-DIMENSIONAL QUASIPERIODIC SYSTEMS: A SCALING APPROACH

1992 ◽  
Vol 06 (03n04) ◽  
pp. 281-320 ◽  
Author(s):  
HISASHI HIRAMOTO ◽  
MAHITO KOHMOTO
Keyword(s):  
Finite Difference ◽  
Finite Number ◽  
Smooth Function ◽  
Wave Functions ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Dense Point ◽  
Quasiperiodic Systems ◽  
Localized Wavefunctions

We review the results of the scaling and multifractal analyses for the spectra and wave-functions of the finite-difference Schrödinger equation: [Formula: see text] Here V is a function of period 1 and ω is irrational. For the Fibonacci model, V takes only two values (it is constant except for discontinuities) and the spectrum is purely singular continuous (critical wavefunctions). When V is a smooth function, the spectrum is purely absolutely continuous (extended wavefunctions) for λ small and purely dense point (localized wavefunctions) for λ large. For an intermediate λ, the spectrum is a mixture of absolutely continuous parts and dense point parts which are separated by a finite number of mobility edges. There is no singular continuous part. (An exception is the Harper model V (x) = cos (2πx), where the spectrum is always pure and the singular continuous one appears at λ = 2.)

Extensions of a renewal theorem

1955 ◽  
Vol 51 (4) ◽  
pp. 629-638 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Random Variables ◽  
Frequency Function ◽  
Renewal Theorem ◽  
Absolutely Continuous ◽  
Image Position ◽  
First Moment ◽  
Independent Identically Distributed

Let X1, X2, X3,… be a sequence of independent, identically distributed, absolutely continuous random variables whose first moment is μ1. Let Sk = X1 + X2 + … + Xk, and let fk(x) be the frequency function of Sk, defined as the k-fold convolution of f1(x). When f1(x) has been defined for all x, fk(x) is uniquely defined for all k, x. Write

scholarly journals Multifractal structure of Bernoulli convolutions

2011 ◽  
Vol 151 (3) ◽  
pp. 521-539 ◽  
Author(s):  
THOMAS JORDAN ◽  
PABLO SHMERKIN ◽  
BORIS SOLOMYAK
Keyword(s):  
Hausdorff Dimension ◽  
Random Variables ◽  
Multifractal Spectrum ◽  
Random Series ◽  
Absolutely Continuous ◽  
Bernoulli Convolutions ◽  
Image Position ◽  
Almost All

AbstractLet νpλbe the distribution of the random series$\sum_{n=1}^\infty i_n \lam^n$, whereinis a sequence of i.i.d. random variables taking the values 0, 1 with probabilitiesp, 1 −p. These measures are the well-known (biased) Bernoulli convolutions.In this paper we study the multifractal spectrum of νpλfor typical λ. Namely, we investigate the size of the setsOur main results highlight the fact that for almost all, and in some cases all, λ in an appropriate range, Δλ,p(α) is nonempty and, moreover, has positive Hausdorff dimension, for many values of α. This happens even in parameter regions for which νpλis typically absolutely continuous.

Steinhaus's geometric location problem for random samples in the plane

1982 ◽  
Vol 14 (1) ◽  
pp. 56-67 ◽  
Author(s):  
Dorit Hochbaum ◽  
J. Michael Steele
Keyword(s):  
Location Problem ◽  
Random Samples ◽  
Recent Developments ◽  
Image Position

Let where Xi, 1 ≦ i ≦ n, are i.i.d. and uniformly distributed in [0, 1]2. It is proved that Mn ∽ cn1–p/2 a.s. for 1 ≦ p <2. This result is motivated by recent developments in the theory of algorithms and the theory of subadditive processes as well as by a well-known problem of H. Steinhaus.

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