Abstract
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 \({{\varvec{a}}}\)-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 \({\varvec{a}}\)-adic solenoid.
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References
Feldman, G.M.: Gaussian distributions on locally compact Abelian groups. Theory Probab. Appl. 23, 529–542 (1979)
Feldman, G.M.: Cauchy distribution on Abelian groups and its characterization. Theory Probab. Appl. 36, 670–681 (1991)
Feldman, G.M.: Arithmetic of probability distributions and characterization problems on Abelian groups. Transl. Math. Monographs. Am. Math. Soc., Providence, RI, 116 (1993)
Feldman, G.M.: On a characterization theorem for the Cauchy distribution on abelian groups. Siberian Math. J. 36, 196–201 (1995)
Feldman, G.M.: A characterization of the Gaussian distribution on Abelian groups. Probab. Theory Relat. Fields 126, 91–102 (2003)
Feldman, G.M.: On a characterization theorem for locally compact abelian groups. Probab. Theory Relat. Fields 133, 345–357 (2005)
Feldman, G.M.: Functional equations and characterization problems on locally compact Abelian groups. EMS Tracts in Mathematics, 5. European Mathematical Society, Zurich (2008)
Feldman, G.M.: Independent linear statistics an \({\varvec {a}}\)-adic solenoids. Theory Probab. Appl. 54, 375–388 (2010)
Feldman, G.M.: The Heyde theorem for locally compact Abelian groups. J. Funct. Anal. 258, 3977–3987 (2010)
Feldman, G.M.: On a characterization of convolutions of Gaussian and Haar distributions. Math. Nachr. 286, 340–348 (2013)
Feldman, G.M.: On the Skitovich-Darmois theorem for the group of p-adic numbers. J. Theor. Probab. 28, 539–549 (2015)
Feldman, G.M.: On a characterization theorem for connected locally compact Abelian groups. J. Fourier Anal. Appl. 26(14), 1–22 (2020)
Feldman, G.M., Graczyk, P.: On the Skitovich-Darmois theorem on compact Abelian groups. J. Theor. Probab. 13, 859–869 (2000)
Feldman, G.M., Graczyk, P.: The Skitovich-Darmois theorem for locally compact Abelian groups. J. Aust. Math. Soc. 88, 339–352 (2010)
Feldman, G.M., Myronyuk, M.V.: Independent linear forms on connected Abelian groups. Math. Nach. 284, 255–265 (2011)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. 1. Grundlehren Math. Wiss. 115, Berlin – Gottingen – Heildelberg (1963)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. 2. Grundlehren Math. Wiss, 152. Springer, New York (1970)
Kagan, A.M., Linnik, Yu. V., Rao, C.R.: Characterization problems in mathematical statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1973)
Myronyuk, M.V.: The Heyde theorem on \({\varvec {a}}\)-adic solenoids. Colloquium Mathematicum. 132, 195–210 (2013)
Myronyuk, M.V.: Independent linear forms on the group \(\Omega _p\). J. Theor. Probab. 33, 1–21 (2020)
Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)
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Feldman, G. Generalized Pólya’s Theorem on a-adic Solenoids. Results Math 76, 124 (2021). https://doi.org/10.1007/s00025-021-01443-0
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DOI: https://doi.org/10.1007/s00025-021-01443-0