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A dichotomy for infinite convolutions of discrete measures

  • G. Brown (a1) and W. Moran (a1)

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Measures, μ which can be realized as an infinite convolution

where each measure μn is a discrete measure, arise naturally in many parts of analysis and number theory (see (15)). The basic property of these measures is ‘purity’; i.e. such a measure μ 1must be absolutely continuous, continuous and singular, or discrete.

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References

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(1)Brown, G. and Moran, W.Translation and power independence for Bernoulli convolutions. Colloq. Math. 27 (1973).
(2)Brown, G. and Moran, W.In general, Bernoulli convolutions have independent powers. Studia Math. 47 (1973).
(3)Ellis, R.Locally compact transformation groups. Duke Math. J. 24 (1957), 119125.
(4)Garsia, A. M.Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409432.
(5)Garsia, A. M.Entropy and singularity of infinite convolutions. Pacific J. Math. 13 (1963), 11591169.
(6)Halmos, P. R.Measure theory (Van Nostrand, Princeton, 1950).
(7)Hewitt, E. and Kakutani, S.Some multiplicative linear functional on M(G). Ann. of Math. 79 (1964), 489505.
(8)Kahane, J. P. and Salem, R.Sur la convolution d'une infinité de distributions de Bernoulli. Colloq. Math. 6 (1958), 193202.
(9)Kahane, J. P. and Salem, R.Ensembles parfaits et séries trigonométriques (Hermann, Paris 1963).
(10)Kaufman, R.Some measures determined by mappings of the Cantor set. Colloq. Math. 19 (1968), 7783.
(11)Pakshirajan, R. P.An analogue of Kolmogorov's Three Series Theorem for abstract random variables. Pacific J. Math. 13 (1963), 639646.
(12)Salem, R.Algebraic numbers and Fourier analysis (Heath, Boston, 1963).
(13)ŠReider, Yu. A.The structure of maximal ideals in rings of measures with convolution. Mat. Sb. 27 (1950), 297318; Amer. Math. Soc. Transl. no. 81.
(14)Taylor, J. L. Inverse, logarithms and idempotents in M(G). (To appear.)
(15)Wintner, A.Lectures on asymptotic distributions and infinite convolutions (Institute for Advanced Study, Princeton, 1938).
(16)Wintner, A.The Fourier transforms of probability distributions (The Johns Hopkins University, Baltimore, 1947).

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