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 Mat. Sb. (N.S.), 1977, Volume 104(146), Number 2(10), Pages 227–247 (Mi msb3876)

This article is cited in 21 scientific papers (total in 21 papers)

On the question of absolute continuity and singularity of probability measures

Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev

Abstract: The basic result in this paper (Theorem 1) generalizes the well-known criterion of Kakutani to measures corresponding to arbitrary random sequences. The proof is based on Theorem 6, which gives a description of the set of convergence of a submartingale with bounded increments. The question of absolute continuity and singularity of measures corresponding to solutions of stochastic difference equations is studied. The dichotomy for Gaussian measures is obtained as a corollary.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 33:2, 203–221

Bibliographic databases:

UDC: 519.2
MSC: Primary 60G30; Secondary 28A35, 60F20, 60G15, 60G45, 60G50, 60J10
Received: 17.03.1977

Citation: Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “On the question of absolute continuity and singularity of probability measures”, Mat. Sb. (N.S.), 104(146):2(10) (1977), 227–247; Math. USSR-Sb., 33:2 (1977), 203–221

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “Absolute continuity and singularity of locally absolutely continuous probability distributions. I”, Math. USSR-Sb., 35:5 (1979), 631–680
2. A. A. Novikov, “On conditions for absolute continuity of probability measures”, Math. USSR-Sb., 35:5 (1979), 697–707
3. Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “Absolute continuity and singularity of locally absolutely continuous probability distributions. II”, Math. USSR-Sb., 36:1 (1980), 31–58
4. Maniia M., “On the Absolute Continuity and Singularity of Probability-Measures of Random-Fields”, 253, no. 3, 1980, 530–533
5. Eagleson G., “An Extended Dichotomy Theorem for Sequences of Pairs of Gaussian Measures”, Ann. Probab., 9:3 (1981), 453–459
6. A. A. Gushchin, “On absolute continuity and singularity of the distributions of random fields”, Math. USSR-Sb., 46:2 (1983), 161–170
7. R. Sh. Liptser, F. Pukel'sheim, A. N. Shiryaev, “Necessary and sufficient conditions for contiguity and entire asymptotic separation of probability measures”, Russian Math. Surveys, 37:6 (1982), 107–136
8. Spataru A., “On Absolute Continuity and Singularity of Product Probability-Measures”, 28, no. 2, 1983, 171–193
9. Vovk V., “On a Criterion of Randomness”, 294, no. 6, 1987, 1298–1302
10. V. I. Bogachev, O. G. Smolyanov, “Analytic properties of infinite-dimensional distributions”, Russian Math. Surveys, 45:3 (1990), 1–104
11. I. A. Dzhvarshejshvili, “On Amarts with Continuous Time”, Theory Probab Appl, 39:3 (1994), 512
12. K Skouras, “Absolute continuity of Markov chains”, Journal of Statistical Planning and Inference, 75:1 (1998), 1
13. A.R. Darwich, “About the absolute continuity and orthogonality for two probability measures”, Statistics & Probability Letters, 52:1 (2001), 1
14. Lars Peter Hansen, Thomas J. Sargent, “Robust estimation and control under commitment”, Journal of Economic Theory, 124:2 (2005), 258
15. Vovk V., “Leading Strategies in Competitive on-Line Prediction”, Algorithmic Learning Theory, Proceedings, Lecture Notes in Artificial Intelligence, 4264, eds. Balcazar J., Long P., Stephan F., Springer-Verlag Berlin, 2006, 214–228
16. Johansson A., Oberg A., Pollicott M., “Countable State Shifts and Uniqueness of G-Measures”, Am. J. Math., 129:6 (2007), 1501–1511
17. Vladimir Vovk, “Leading strategies in competitive on-line prediction”, Theoretical Computer Science, 405:3 (2008), 285
18. Akio Fujiwara, “Randomness Criteria in Terms of <formula formulatype="inline"><tex>${\alpha }$</tex></formula>-Divergences”, IEEE Trans. Inform. Theory, 54:3 (2008), 1252
19. Valerii V. Buldygin, Marina K. Runovska, “On the convergence of series of autoregressive sequences in Banach spaces”, Theory Stoch. Process., 16(32):1 (2010), 29–38
20. S. S. Gabriyelyan, “Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space”, J Theor Probab, 2011
21. A. A. Gushchin, “The joint law of terminal values of a nonnegative submartingale and its compensator”, Theory Probab. Appl., 62:2 (2018), 216–235
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