A Generalized Condition for Random Vectors to be Identically Distributed Related to the Analytical Theory of Linear Forms of Independent Random Variables

1990 ◽  
Vol 34 (2) ◽  
pp. 327-332 ◽  
Author(s):  
A. M. Kagan
Keyword(s):  
Random Variables ◽  
Analytical Theory ◽  
Independent Random Variables ◽  
Random Vectors ◽  
Linear Forms

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 Supermodular ordering of Poisson and binomial random vectors by tree-based correlations

2018 ◽  
Vol 38 (2) ◽  
pp. 385-405
Author(s):  
Nicolas Privault ◽  
Bünyamin Kizildemir
Keyword(s):  
Random Variables ◽  
Covariance Matrices ◽  
Binary Trees ◽  
Independent Random Variables ◽  
Dependence Structure ◽  
Random Vectors ◽  
Partially Ordered ◽  
Ordered Trees ◽  
Lévy Measures ◽  
Inversion Techniques

We construct a dependence structure for binomial, Poisson and Gaussian random vectors, based on partially ordered binary trees and sums of independent random variables. Using this construction, we characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. For this, we apply Möbius inversion techniques on partially ordered trees, which allow us to connect the Lévy measures of Poisson random vectors on the discrete d-dimensional hypercube to their covariance matrices.

On the Skitovich-Darmois-Ramachandran-Ibragimov theorem for linear forms of Q-independent random sequences

2018 ◽  
Vol 55 (3) ◽  
pp. 353-362
Author(s):  
B. L. S. Prakasa Rao
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Linear Forms ◽  
Random Sequences ◽  
Infinite Sequences

We characterize the normal distribution based on the Q-independence of linear forms based on infinite sequences of Q-independent random variables.

scholarly journals THE SKITOVICH–DARMOIS THEOREM FOR LOCALLY COMPACT ABELIAN GROUPS

2010 ◽  
Vol 88 (3) ◽  
pp. 339-352 ◽  
Author(s):  
GENNADIY FELDMAN ◽  
PIOTR GRACZYK
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Wide Class ◽  
Random Variables ◽  
Independent Random Variables ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Forms ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

AbstractAccording to the Skitovich–Darmois theorem, the independence of two linear forms of n independent random variables implies that the random variables are Gaussian. We consider the case where independent random variables take values in a second countable locally compact abelian group X, and coefficients of the forms are topological automorphisms of X. We describe a wide class of groups X for which a group-theoretic analogue of the Skitovich–Darmois theorem holds true when n=2.

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