On a characterization of convolutions of Gaussian and Haar distributions

Feldman, G. M.

doi:10.1002/mana.201100320

Cited by 25 publications

(15 citation statements)

References 19 publications

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“…It follows from this that f (4y) = (f (y)) 9 (φ(2y)) 3 φ(4y), and we may also suppose that m = 4 in (11).…”

Section: Main Theoremmentioning

confidence: 99%

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On a Characterization Theorem for Connected Locally Compact Abelian Groups

Feldman

2020

J Fourier Anal Appl

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According to the generalized Pólya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a complete description of a-adic solenoids for which an analog of this theorem is true. The proof of the main theorem is reduced to solving some functional equation in the class of continuous positive definite functions on the character group of an a-adic solenoid.

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“…It follows from this that f (4y) = (f (y)) 9 (φ(2y)) 3 φ(4y), and we may also suppose that m = 4 in (11).…”

Section: Main Theoremmentioning

confidence: 99%

“…A number of papers have been devoted to generalizing of these theorems to various algebraic structures, in particular, to locally compact Abelian groups (see e.g. [5,6,[8][9][10][11][12][13][14]18], and also [7,). In so doing, the coefficients of linear forms are topological automorphisms of the group.…”

Section: Introductionmentioning

confidence: 99%

On a Characterization Theorem for Connected Locally Compact Abelian Groups

Feldman

2020

J Fourier Anal Appl

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“…The case when random variables take values in locally compact Abelian groups was studied in details (see, e.g., [1][2][3][4][5][6][7][8], ).…”

Section: Introductionmentioning

confidence: 99%

“…In [8], G.M. Feldman studied the problem of the chracterization of convolutions of Gaussian and Haar distributions on an arbitrary locally compact Abelian group.…”

Section: Introductionmentioning

confidence: 99%

On a Characterization of the Haar Distribution on Compact Abelian Groups

Mazur¹

2014

Z. mat. fiz. anal. geom.

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“…Assume that the left-hand side in (10) is not equal to zero. Then (9) implies that u + v, u + αv ∈ Z(p k 1 ). It follows from this that…”

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confidence: 99%

The Heyde characterization theorem on compact totally disconnected and connected Abelian groups

Feldman

2021

Preprint

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By the well-known 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 another. In the case of two independent random variables we give a complete description of compact totally disconnected Abelian groups X, where an analogue of this theorem is valid. We also prove that even a weak analogue of the Heyde theorem fails on compact connected Abelian groups X. Coefficients of considered linear forms are topological automorphisms of X. The proofs are based on the study of solutions of a functional equation on the character group of the group X in the class of Fourier transforms of probability distributions.

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On a characterization of convolutions of Gaussian and Haar distributions

Cited by 25 publications

References 19 publications

On a Characterization Theorem for Connected Locally Compact Abelian Groups

On a Characterization Theorem for Connected Locally Compact Abelian Groups

On a Characterization of the Haar Distribution on Compact Abelian Groups

The Heyde characterization theorem on compact totally disconnected and connected Abelian groups

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