A measure which is singular and uniformly locally uniform
HTML articles powered by AMS MathViewer
- by David Freedman and Jim Pitman
- Proc. Amer. Math. Soc. 108 (1990), 371-381
- DOI: https://doi.org/10.1090/S0002-9939-1990-0990427-5
- PDF | Request permission
Abstract:
An example is given of a singular measure on $[0,1]$ which is locally nearly uniform in the weak star topology. If this measure is used as a prior to estimate an unknown probability in coin tossing, the posterior is asymptotically normal.References
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578 P. S. Laplace, Mémoire sur les intégrales définies et leur application aux probabilités, et spéciale ment à la recherche du milieu qu’il faut choisir entre les résultats des observations, Mémoires présentés à l’Académie des Sciences, Paris, 1809. S. Bernstein, Theory of probability (in Russian), Moscow, 1917. R. von Mises, Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin, (1931).
- Lucien Le Cam, Asymptotic methods in statistical decision theory, Springer Series in Statistics, Springer-Verlag, New York, 1986. MR 856411, DOI 10.1007/978-1-4612-4946-7 —, On the Bernstein-von Mises theorem, Technical report no. 57, Department of Statistics, University of California, Berkeley, (1986).
- Antoni Zygmund, On lacunary trigonometric series, Trans. Amer. Math. Soc. 34 (1932), no. 3, 435–446. MR 1501647, DOI 10.1090/S0002-9947-1932-1501647-3 —, Trigonometric Series, two volumes, The University Press, Cambridge, 1959. P. Diaconis and D. Freedman, On the uniform consistency of Bayes estimates for multinomial probabilities, Technical report no. 137, Department of Statistics, University of California, Berkeley, 1988.
- Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. MR 0422992 J. Peyière, Sur les produits de Riesz, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), 1453-1455.
- Gavin Brown and William Moran, On orthogonality of Riesz products, Proc. Cambridge Philos. Soc. 76 (1974), 173–181. MR 350319, DOI 10.1017/s0305004100048830
- Gavin Brown and William Moran, Coin tossing and powers of singular measures, Math. Proc. Cambridge Philos. Soc. 77 (1975), 349–364. MR 358906, DOI 10.1017/S0305004100051173
- Gavin Brown, Singular infinitely divisible distributions whose characteristic functions vanish at infinity, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 2, 277–287. MR 442600, DOI 10.1017/S0305004100053913
- Friedrich Riesz, Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung, Math. Z. 2 (1918), no. 3-4, 312–315 (German). MR 1544321, DOI 10.1007/BF01199414
- Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606 D. Freedman and J. W. Pitman, A singular measure which is locally uniform, Technical report no. 163, Department of Statistics, University of California, Berkeley, 1988.
- Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 23331, DOI 10.2307/1969123
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 371-381
- MSC: Primary 28A12; Secondary 62F12, 62F15
- DOI: https://doi.org/10.1090/S0002-9939-1990-0990427-5
- MathSciNet review: 990427