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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
\by Yu.~M.~Kabanov, R.~Sh.~Liptser, A.~N.~Shiryaev
\paper On the question of absolute continuity and singularity of probability measures
\jour Mat. Sb. (N.S.)
\yr 1977
\vol 104(146)
\issue 2(10)
\pages 227--247
\jour Math. USSR-Sb.
\yr 1977
\vol 33
\issue 2
\pages 203--221

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    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  mathnet  crossref  mathscinet  zmath  isi
    2. A. A. Novikov, “On conditions for absolute continuity of probability measures”, Math. USSR-Sb., 35:5 (1979), 697–707  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  isi
    4. Maniia M., “On the Absolute Continuity and Singularity of Probability-Measures of Random-Fields”, 253, no. 3, 1980, 530–533  mathscinet  isi
    5. Eagleson G., “An Extended Dichotomy Theorem for Sequences of Pairs of Gaussian Measures”, Ann. Probab., 9:3 (1981), 453–459  crossref  mathscinet  zmath  isi
    6. A. A. Gushchin, “On absolute continuity and singularity of the distributions of random fields”, Math. USSR-Sb., 46:2 (1983), 161–170  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. Spataru A., “On Absolute Continuity and Singularity of Product Probability-Measures”, 28, no. 2, 1983, 171–193  mathscinet  isi
    9. Vovk V., “On a Criterion of Randomness”, 294, no. 6, 1987, 1298–1302  mathscinet  zmath  isi
    10. V. I. Bogachev, O. G. Smolyanov, “Analytic properties of infinite-dimensional distributions”, Russian Math. Surveys, 45:3 (1990), 1–104  mathnet  crossref  mathscinet  zmath  isi
    11. I. A. Dzhvarshejshvili, “On Amarts with Continuous Time”, Theory Probab Appl, 39:3 (1994), 512  mathnet  crossref  mathscinet  zmath  isi
    12. K Skouras, “Absolute continuity of Markov chains”, Journal of Statistical Planning and Inference, 75:1 (1998), 1  crossref  mathscinet  zmath
    13. A.R. Darwich, “About the absolute continuity and orthogonality for two probability measures”, Statistics & Probability Letters, 52:1 (2001), 1  crossref  mathscinet  zmath
    14. Lars Peter Hansen, Thomas J. Sargent, “Robust estimation and control under commitment”, Journal of Economic Theory, 124:2 (2005), 258  crossref  mathscinet  zmath
    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  mathscinet  zmath  isi
    16. Johansson A., Oberg A., Pollicott M., “Countable State Shifts and Uniqueness of G-Measures”, Am. J. Math., 129:6 (2007), 1501–1511  crossref  mathscinet  zmath  isi
    17. Vladimir Vovk, “Leading strategies in competitive on-line prediction”, Theoretical Computer Science, 405:3 (2008), 285  crossref  mathscinet  zmath
    18. Akio Fujiwara, “Randomness Criteria in Terms of <formula formulatype="inline"><tex>$ {\alpha }$</tex></formula>-Divergences”, IEEE Trans. Inform. Theory, 54:3 (2008), 1252  crossref  mathscinet  zmath
    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  mathnet  mathscinet  zmath
    20. S. S. Gabriyelyan, “Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space”, J Theor Probab, 2011  crossref  mathscinet  zmath
    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  mathnet  crossref  crossref  mathscinet  isi  elib
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