Minimal metrics in the real random variables space

1983 ◽  
pp. 172-190 ◽  
Author(s):  
S. T. Rachev
Keyword(s):  
Random Variables ◽  
The Real ◽  
Minimal Metrics

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.

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.

scholarly journals On the real representation of quaternion random variables

Statistics ◽  
2013 ◽  
Vol 47 (6) ◽  
pp. 1224-1240 ◽  
Author(s):  
Mattheüs T. Loots ◽  
Andriëtte Bekker ◽  
Mohammad Arashi ◽  
Jacobus J.J. Roux
Keyword(s):  
Random Variables ◽  
Real Representation ◽  
The Real

scholarly journals Random Variables and Product of Probability Spaces

2013 ◽  
Vol 21 (1) ◽  
pp. 33-39
Author(s):  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Borel Sets ◽  
The Real ◽  
Probability Spaces ◽  
Countably Infinite

Summary We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of countably infinite probability spaces.

scholarly journals Identification of the theory of orthogonal polynomials in d-indeterminates with the theory of 3-diagonal symmetric interacting Fock spaces on ℂd

2017 ◽  
Vol 20 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Luigi Accardi ◽  
Abdessatar Barhoumi ◽  
Ameur Dhahri
Keyword(s):  
Orthogonal Polynomials ◽  
Polynomial Algebra ◽  
Random Variables ◽  
Positive Definite ◽  
Probability Measures ◽  
Fock Spaces ◽  
Positive Definite Kernels ◽  
Algebraic Classification ◽  
The Real

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on [Formula: see text] with moments of any order and more generally of states on the polynomial algebra on [Formula: see text]. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

STOCHASTIC EQUIVALENCE OF CONVEX ORDERED DISTRIBUTIONS AND APPLICATIONS

2000 ◽  
Vol 14 (1) ◽  
pp. 33-48 ◽  
Author(s):  
M. C. Bhattacharjee ◽  
R. N. Bhattacharya
Keyword(s):  
Recent Result ◽  
Real Line ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Survival Times ◽  
The Real ◽  
Ordered Alternatives ◽  
Special Cases ◽  
Stochastic Equivalence

We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.

scholarly journals The 123 theorem of Probability Theory and Copositive Matrices

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Alexander Kovačec ◽  
Miguel M. R. Moreira ◽  
David P. Martins
Keyword(s):  
Probability Theory ◽  
Integral Inequality ◽  
Random Variables ◽  
Monotone Functions ◽  
The Real ◽  
Copositive Matrices ◽  
Vector Valued ◽  
Independent Identically Distributed

AbstractAlon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.

scholarly journals Probability on Finite Set and Real-Valued Random Variables

2009 ◽  
Vol 17 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Random Variables ◽  
Theoretical Frameworks ◽  
The Real ◽  
Formalized Mathematics ◽  
Finite Set

Probability on Finite Set and Real-Valued Random Variables In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.

A one dimensional random space-filling problem

1968 ◽  
Vol 5 (2) ◽  
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.

scholarly journals Separation metrics for real-valued random variables

1984 ◽  
Vol 7 (2) ◽  
pp. 407-408
Author(s):  
Michael D. Taylor
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
The Real ◽  
Work Done

IfWis a fixed, real-valued random variable, then there are simple and easily satisfied conditions under which the functiondW, wheredW(X,Y)=the probability thatW“separates” the real-valued random variablesXandY, turns out to be a metric. The observation was suggested by work done in [1].

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