A central limit theorem for exchangeable random variables on a locally compact abelian group

2000 ◽  
Vol 74 (2) ◽  
pp. 115-119
Author(s):  
M. S. Bingham
Keyword(s):  
Abelian Group ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Compact Abelian Group ◽  
Random Variables ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Exchangeable Random Variables

scholarly journals On C.R. RAO’s theorem for locally compact abelian groups

2019 ◽  
Vol 484 (3) ◽  
pp. 273-276
Author(s):  
G. M. Feldman
Keyword(s):  
Abelian Group ◽  
Random Vector ◽  
Compact Abelian Group ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let x1, x2, x3 be independent random variables with values in a locally compact Abelian group X with nonvanish- ing characteristic functions, and aj, bj be continuous endomorphisms of X satisfying some restrictions. Let L1 = a1x1 + a2x2 + a3x3, L2 = b1x1 + b2x2 + b3x3. It was proved that the distribution of the random vector (L1; L2) determines the distributions of the random variables xj up a shift. This result is a group analogue of the well-known C.R. Rao theorem. We also prove an analogue of another C.R. Rao’s theorem for independent random variables with values in an a-adic solenoid.

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.

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

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