Abstract
Let X be a real-valued random variable and \(M\) a σ-algebra. We show that the minimum \({\mathbb{L}}^1\)-distance between X and a random variable distributed as X and independant of \(M\) can be viewed as a dependence coefficient τ(\(M\),X) whose definition is comparable (but different) to that of the usual β-mixing coefficient between \(M\) and σ(X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of τ-dependent sequences. The former is used to prove a strong invariance principle for partial sums.
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REFERENCES
Andrews, D. W. K. (1984). Nonstrong mixing autoregressive processes. J.Appl.Probab. 21, 930–934.
Berbee, H. C. P. (1979). Random walks with stationary increments and renewal theory. Math.Cent.Tracts. Amsterdam.
Billingsley, P. (1968). Convergence of probability measures. Wiley, New York.
Billingsley, P. (1995). Probability and Measure, 3rd ed., Wiley, New York.
Bosq, D. (1993). Bernstein 's type large deviation inequalities for partial sums of strong mixing process. Statistics 24, 59–70.
Bradley, R. C. (1983). Approximations theorems for strongly mixing random variables. Michigan Math.J. 30, 69–81.
Bradley, R. C. (2002). Introduction to Strong Mixing Conditions, Vol 1. Technical Report, Department of Mathematics, I. U. Bloomington.
Broise, A. (1996). Transformations dilatantes de 1 'intervalle et th ´eorèmes limites. ´Etudes spectrales d 'op ´erateurs de transfert et applications. Ast ´erisque 238, 1–109.
Bryc, W. (1982). On the approximation theorem of Berkes and Philipp, Demonstratio Mathematica 15, 807–816.
Coulon-Prieur, C., and Doukhan, P. (2000). A triangular CLT for weakly dependent sequences. Statist.Probab.Lett. 47, 61–68.
Dedecker, J., and Doukhan, P. (2002). A new covariance inequality and applications, Stochastic Process.Appl. 106, 63–80.
Dedecker, J., and Rio, E. (2000). On the functional central limit theorem for stationary processes. Ann.Inst.H.Poincar ´e Probab.Statist. 36, 1–34.
Dehling, H., and Philipp, W. (1982). Almost sure invariance principles for weakly dependent vector-valued random variables, Ann.Probab. 10, 689–701.
Doukhan, P. (1994). Mixing:properties and examples. Lecture Notes in Statist. 85, Springer-Verlag.
Doukhan, P., and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process.Appl. 84, 313–342.
Doukhan, P., Massart P., and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann.Inst.H.Poincar ´e Probab.Statist. 30, 63–82.
Dudley, R. M., (1989). Real analysis and probability. Wadworsth Inc., Belmont, California.
Fuk, D. K., and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Probab.Appl. 16, 643–660.
Goldstein, S. (1979). Maximal coupling. Z.Wahrsch.Verw.Gebiete 46, 193–204.
Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl.Akad.Nauk SSSR 188, 739–741.
Gordin, M. I. (1973). Abstracts of Communication, T. 1:A-K, International Conference on Probability Theory, Vilnius.
Heyde, C. C., and Scott, D. J. (1973). Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments. Ann.Probab. 1, 428–436.
Heyde, C. C. (1975). On the central limit theorem and iterated logarithm law for stationary processes. Bull.Austral.Math.Soc. 12, 1–8.
Major, P. (1978). On the invariance principle for sums of identically distributed random variables. J.Multivariate Anal. 8, 487–517.
Merlevède, F., and Peligrad, M. (2002). On the coupling of dependent random variables and applications. Empirical process techniques for dependent data. 171–193. Birkh ¨auser.
McShane, E. J. (1947). Integration. Princeton university press, London.
Oodaira, H., and Yoshihara, K. I. (1971). Note on the law of the iterated logarithm for stationary processes satisfying mixing conditions. Kodai Math.Sem.Rep. 23, 335–342.
Peligrad, M. (2001). A note on the uniform laws for dependent processes via coupling. J.Theoret.Probab. 14, 979–988.
Peligrad, M. (2002). Some remarks on coupling of dependent random variables. Stat.Prob.Letters 60, 201–209.
Philipp, W. (1986). Invariance principles for independent and weakly dependent random variables. Dependence in probability and statistics (Oberwolfach,1985), 225–268. Prog. Probab. Statist.11, Birkh ¨auser, Boston.
Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann.Probab. 23, 1188–1203.
Rio, E. (2000). Th ´eorie asymptotique des processus al ´eatoires faiblement d ´ependants. Collection Math ´ematiques & Applications 31. Springer, Berlin.
Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition, Proc.Nat.Acad.Sci.U.S.A. 42, 43–47.
Shao, Q-. M. (1993). Almost sure invariance principle for mixing sequences of random variables. Stochastic Process.Appl. 48, 319–334.
Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z.Wahrsch.Verw.Gebiete 3, 211–226.
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Dedecker, J., Prieur, C. Coupling for τ-Dependent Sequences and Applications. Journal of Theoretical Probability 17, 861–885 (2004). https://doi.org/10.1007/s10959-004-0578-x
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DOI: https://doi.org/10.1007/s10959-004-0578-x