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On a characterization theorem on \(a\)-adic solenoids. (English. Russian original) Zbl 1448.60013

Dokl. Math. 100, No. 3, 538-541 (2019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 489, No. 3, 227-231 (2019).
Summary: According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in independent random variables given another. We prove an analogue of this theorem for linear forms in two independent random variables with values in an \(a\)-adic solenoind without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of the \(a\)-adic solenoid.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
62E10 Characterization and structure theory of statistical distributions
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References:

[1] Heyde, C. C., Sankhya Ser. A, 32, 115-118 (1970) · Zbl 0209.50702
[2] A. M. Kagan, Yu. V. Linnik, and S. R. Rao, Characterization Problems in Mathematical Statistics (Nauka, Moscow, 1972; Wiley, New York, 1973). · Zbl 0243.62009
[3] Feldman, G. M., J. Theor. Probab., 17, 929-941 (2004) · Zbl 1070.60020 · doi:10.1007/s10959-004-0583-0
[4] Mironyuk, M. V.; Feldman, G. M., Sib. Math. J., 46, 315-324 (2005) · Zbl 1100.60003 · doi:10.1007/s11202-005-0033-y
[5] Feldman, G. M., Probab. Theory Relat. Fields, 133, 345-357 (2005) · Zbl 1086.62010 · doi:10.1007/s00440-005-0429-4
[6] Feldman, G. M., Stud. Math., 177, 67-79 (2006) · Zbl 1111.62013 · doi:10.4064/sm177-1-5
[7] Feldman, G. M., J. Funct. Anal., 258, 3977-3987 (2010) · Zbl 1195.60010 · doi:10.1016/j.jfa.2010.03.005
[8] Feldman, G. M., Math. Nachr., 286, 340-348 (2013) · Zbl 1263.60003 · doi:10.1002/mana.201100320
[9] Myronyuk, M. V., Colloq. Math., 132, 195-210 (2013) · Zbl 1285.60005 · doi:10.4064/cm132-2-3
[10] Feldman, G. M., Publ. Math. Debrecen, 87, 147-166 (2015) · Zbl 1363.62009 · doi:10.5486/PMD.2015.7100
[11] Feldman, G. M., Theory Probab. Appl., 62, 399-412 (2018) · Zbl 1410.60009 · doi:10.1137/S0040585X97T988708
[12] G. M. Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts in Mathematics (Eur. Math. Soc., Zurich, 2008), Vol. 5. · Zbl 1156.60003
[13] Parthasarathy, K. R., Probability Measures on Metric Spaces (1967), New York: Academic, New York · Zbl 0153.19101
[14] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis (1970), Berlin: Springer-Verlag, Berlin · Zbl 0213.40103
[15] Feldman, G. M., Probab. Theory Relat. Fields, 126, 91-102 (2003) · Zbl 1016.62005 · doi:10.1007/s00440-003-0256-4
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