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Conditional independence, conditional mixing and conditional association. (English) Zbl 1314.60054
Summary: Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.

60E05 Probability distributions: general theory
60E15 Inequalities; stochastic orderings
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
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[1] Basawa I.V., Prakasa Rao B.L.S. (1980). Statistical inference for stochastic processes. London, Academic · Zbl 0448.62070
[2] Basawa I.V., Scott D. (1983). Asymptotic optimal inference for non-ergodic models. Lecture Notes in Statistics, Vol. 17. New York, Springer · Zbl 0519.62039
[3] Billingsley P. (1986). Probability and measure. New York, Wiley · Zbl 0649.60001
[4] Chow Y.S., Teicher H. (1978). Probability theory: independence, interchangeability, martingales. New York, Springer · Zbl 0399.60001
[5] Chung K.L., Erdös P. (1952). On the application of the Borel-Cantelli lemma. Transactions of American Mathematical Society 72, 179–186 · Zbl 0046.35203
[6] Doob J.L. (1953). Stochastic processes. New York, Wiley · Zbl 0053.26802
[7] Guttorp P. (1991). Statistical inference for branching processes. New York, Wiley · Zbl 0778.62077
[8] Gyires B. (1981). Linear forms in random variables defined on a homogeneous Markov chain. In: Revesz P. et al. (eds). The first pannonian symposium on mathematical statistics, Lecture Notes in Statistics, Vol. 8. New York, Springer, pp. 110–121
[9] Hájek J., Rényi A. (1955). Generalization of an inequality of Kolmogorov. Acta Mathematica Academiae Scientiarum Hungaricae 6, 281–283 · Zbl 0067.10701
[10] Kochen S., Stone C. (1964). A note on the Borel-Cantelli lemma. Illinois Journal of Mathematics 8, 248–251 · Zbl 0139.35401
[11] Loève M. (1977). Probability theory I (4th Ed.). New York, Springer · Zbl 0435.30027
[12] Majerak D., Nowak W., Zieba W. (2005). Conditional strong law of large number. International Journal of Pure and Applied Mathematics 20, 143–157
[13] Newman C. (1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong Y.L. (ed). Inequalities in statistics and probability. Hayward, IMS, pp. 127–140
[14] Petrov V.V. (2004). A generalization of the Borel-Cantelli lemma. Statistics and Probabability Letters 67, 233–239 · Zbl 1101.60300
[15] Prakasa Rao B.L.S. (1987). Characterization of probability measures by linear functions defined on a homogeneous Markov chain. Sankhyā, Series A 49: 199–206 · Zbl 0654.62018
[16] Prakasa Rao B.L.S. (1990). On mixing for flows of {\(\sigma\)}-algebras, Sankhyā. Series A 52: 1–15 · Zbl 0719.60038
[17] Prakasa Rao B.L.S. (1999a). Statistical inference for diffusion type processes. London, Arnold and New York, Oxford University Press · Zbl 0952.62077
[18] Prakasa Rao B.L.S. (1999b). Semimartingales and their statistical inference. Boca Raton, CRC Press · Zbl 0960.62090
[19] Prakasa Rao B.L.S., Dewan I. (2001). Associated sequences and related inference problems. In: Rao C.R., Shanbag D.N. (eds). Handbook of statistics, 19, stochastic Processes: theory and methods. Amsterdam, North Holland, pp. 693–728 · Zbl 1036.62049
[20] Rosenblatt M. (1956). A central limit theorem and a strong mixing condition. Proceedings National Academy of Sciences U.S.A 42: 43–47 · Zbl 0070.13804
[21] Roussas G.G. (1999). Positive and negative dependence with some statistical applications. In Ghosh S. (ed). Asymptotics, nonparametrics and time series. New York, Marcel Dekker, pp. 757–788 · Zbl 1069.62518
[22] Roussas G.G., Ioannides D. (1987). Moment inequalities for mixing sequences of random variables. Stochastic Analysis and Applications 5, 61–120 · Zbl 0619.60022
[23] Yan, J.A. (2004). A new proof of a generalized Borel-Cantelli lemma (Preprint), Academy of Mathematics and Systems Science. Chinese Academy of Sciences, Beijing.
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