scholarly journals On a characterization theorem on a-adic solenoids

2019 ◽  
Vol 489 (3) ◽  
pp. 227-231
Author(s):  
G. M. Feldman
Keyword(s):  
Gaussian Distribution ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
The Other ◽  
Independent Random Variables ◽  
Linear Forms ◽  
The Real

According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We prove an analogue of this theorem for linear forms of two independent random variables taking values in an -adic solenoid containing no elements of order 2. Coefficients of the linear forms are topological automorphisms of the -adic solenoid.

scholarly journals HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 93-102 ◽  
Author(s):  
MARGARYTA MYRONYUK
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
Gaussian Distributions ◽  
Discrete Abelian Group

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

Sensor allocation problems on the real line

2016 ◽  
Vol 53 (3) ◽  
pp. 667-687 ◽  
Author(s):  
Evangelos Kranakis ◽  
Gennady Shaikhet
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Real Line ◽  
Queueing Theory ◽  
Random Variables ◽  
The Other ◽  
Independent Random Variables ◽  
Coverage Problem ◽  
The Real ◽  
Allocation Problems

AbstractA large numbernof sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional ton. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs asnincreases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.

scholarly journals On a group analogue of the Heyde theorem

2020 ◽  
Vol 32 (2) ◽  
pp. 307-318 ◽  
Author(s):  
Margaryta Myronyuk
Keyword(s):  
Abelian Group ◽  
Present Article ◽  
Gaussian Distribution ◽  
Compact Abelian Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Forms

AbstractHeyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analogue of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

On a problem concerning quaternion valued Gaussian random variables

2010 ◽  
Vol 17 (4) ◽  
pp. 629-634
Author(s):  
George Chelidze ◽  
Nicholas Vakhania
Keyword(s):  
Linear Form ◽  
Random Variables ◽  
Characterization Theorem ◽  
Random Variable ◽  
Negative Answer ◽  
The Other ◽  
Gaussian Random Variables ◽  
Gaussian Distributions ◽  
Left Hand ◽  
The Right

Abstract In the present paper we construct an example of a quaternion random variable such that Polya's type characterization theorem of Gaussian distributions does not hold. The matter is that in the linear form, consisting of the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and the other part on the left-hand side. This gives a negative answer to the question posed in [Vakhania and Chelidze, Teor. Veroyatnost. i Primenen. 54: 337–344, 2009].

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

On indirect Methods of acquiring Knowledge - The Method of History. - The First Table of Mortality

1851 ◽  
Vol 1 (1) ◽  
pp. 40-46
Author(s):  
Edwin James Farren
Keyword(s):  
Real Line ◽  
English Language ◽  
The Other ◽  
Adult Student ◽  
Indirect Methods ◽  
The Real ◽  
Straight Line ◽  
Point To Point ◽  
The One

The term scholar, as current in the English language, has two extreme acceptations, tyro and proficient; or what the later Greeks fancifully termed the alpha and omega of acquirement. If we attempt to trace the steps by which even the adult student of any especial branch of professional or literary knowledge has fairly passed the boundary defined by the one meaning in passing on to that position denoted by the other, it will commonly be found, that in place of that lucid order, that straight line from point to point, which theory and resolve generally premise, the real order of acquirement has been desultory—the real line of progression, circuitous and uncertain.

A characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (3) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

scholarly journals The product of independent random variables with slowly varying truncated moments

1997 ◽  
Vol 62 (2) ◽  
pp. 186-197 ◽  
Author(s):  
Takaaki Shimura
Keyword(s):  
Gaussian Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Decomposition Of Distributions

AbstractThe Mellin-Stieltjes convolution and related decomposition of distributions in M(α) (the class of distributions μ on (0, ∞) with slowly varying αth truncated moments ) are investigated. Maller shows that if X and Y are independent non-negative random variables with distributions μ and v, respectively, and both μ and v are in D2, the domain attraction of Gaussian distribution, then the distribution of the product XY (that is, the Mellin-Stieltjes convolution μ ^ v of μ and v) also belongs to it. He conjectures that, conversely, if μ ∘ v belongs to D2, then both μ and v are in it. It is shown that this conjecture is not true: there exist distributions μ ∈ D2 and v μ ∈ D2 such that μ ^ v belongs to D2. Some subclasses of D2 are given with the property that if μ ^ v belongs to it, then both μ and v are in D2.

A one dimensional random space-filling problem

1968 ◽  
Vol 5 (02) ◽  
pp. 427-435 ◽  
Author(s):  
John P. Mullooly
Keyword(s):  
Asymptotic Behavior ◽  
Finite Number ◽  
Real Line ◽  
Random Variables ◽  
Random Interval ◽  
Space Filling ◽  
One Dimensional ◽  
The Real ◽  
Fixed Constant ◽  
Filling Problem

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx , the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

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