scholarly journals Characterization of distributions of Q-independent random variables on locally compact Abelian groups

2019 ◽  
Vol 152 ◽  
pp. 82-88 ◽  
Author(s):  
Margaryta Myronyuk
Keyword(s):  
Abelian Groups ◽  
Random Variables ◽  
Independent Random Variables ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Characterization Of Distributions ◽  
Compact Abelian Groups

Scaling sets and generalized scaling sets on Cantor dyadic group

2020 ◽  
Vol 18 (04) ◽  
pp. 2050019
Author(s):  
Prasadini Mahapatra ◽  
Divya Singh
Keyword(s):  
Abelian Groups ◽  
Real Case ◽  
Locally Compact ◽  
Wavelet Sets ◽  
Locally Compact Abelian Groups ◽  
Dyadic Group ◽  
Compact Abelian Groups ◽  
Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Pego everywhere

2016 ◽  
Vol 15 (04) ◽  
pp. 1650074 ◽  
Author(s):  
Przemysław Górka ◽  
Tomasz Kostrzewa
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
General Version ◽  
Pontryagin Duality ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform ◽  
Compact Subsets

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.

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.

PEGO THEOREM ON LOCALLY COMPACT ABELIAN GROUPS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350143 ◽  
Author(s):  
PRZEMYSłAW GÓRKA
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform

In this paper, we show the version of Pego's theorem on locally compact abelian groups. This theorem, [R. L. Pego, Compactness in L2 and the Fourier transform, Proc. Amer. Math. Soc.95 (1985) 252–254], gives a characterization of precompact sets of L2 in terms of the Fourier transform.

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