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 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.

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

scholarly journals Jackson's Theorem for locally compact abelian groups

1974 ◽  
Vol 10 (1) ◽  
pp. 59-66 ◽  
Author(s):  
Walter R. Bloom
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Integrable Function ◽  
Locally Compact Abelian Group ◽  
Classical Theorem ◽  
Extended Version ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Approximate Unit ◽  
Compact Abelian Groups

If f is a p–th integrable function on the circle group and ω(p; f; δ) is its mean modulus of continuity with exponent p then an extended version of the classical theorem of Jackson states the for each positive integer n, there exists a trigonometric polynomial tn of degree at most n for which‖f-tn‖p ≤(p; f; 1/n).In this paper it will be shewn that for G a Hausdorff locally compact abelian group, the algebra L1(G) admits a certain bounded positive approximate unit which, in turn, will be used to prove an analogue of the above result for Lp(G).

On the Upper Majorant Property for Locally Compact Abelian Groups

1978 ◽  
Vol 30 (5) ◽  
pp. 915-925
Author(s):  
M. Rains
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let G be a compact abelian group and form the spaces LP(G) with respect to the normalized Haar measure on G.

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.

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