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Large deviations for the empirical mean of associated random variables. (English) Zbl 1151.60013
Let $$(X_n)_{n\geq1}$$ be a strictly stationary sequence of associated random variables which are uniformly bounded. Under the assumption of an hyper-geometric decay of $$\sum_{j=n}^\infty\text{Cov}(X_1,X_j)$$ and a boundedness assumption on the densities of the sample means $$\overline{X}_n=n^{-1}\sum_{i=1}^nX_i$$ it is shown that the sequence $$(\overline{X}_n)_{n\geq1}$$ satisfies the large deviation principle.

##### MSC:
 60F10 Large deviations 60G15 Gaussian processes
##### Keywords:
large deviations; association; stationarity
Full Text:
##### References:
 [1] Bryc, W., On large deviations for uniformly strong mixing sequences, Stoch. processes appl., 41, 191-202, (1992) · Zbl 0756.60027 [2] Bryc, W.; Dembo, A., Large deviations and strong mixing, Ann. inst. Henri Poincaré, 32, 549-569, (1996) · Zbl 0863.60028 [3] Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, Springer-Verlag, New York. · Zbl 0896.60013 [4] Dewan, I., Prakasa Rao, B.L.S., 2001. Associated sequences and related inference problems, In: Shanbhag, D.N., Rao, C.R. (Eds.), Stochastic Processes: Theory and Methods. Handbook of Statistics 19, North-Holland, Amsterdam, pp. 693-731. · Zbl 1036.62049 [5] Esary, J.D.; Proschan, F.; Walkup, D.W., Association of random variables with applications, Ann. math. statist., 38, 1466-1474, (1967) · Zbl 0183.21502 [6] Fortuin, C.M.; Kasteleyn, P.W.; Ginibre, J., Correlation inequalities on some partially ordered sets, Commun. math. phys., 22, 89-103, (1971) · Zbl 0346.06011 [7] Newman, C.M., Normal fluctuations and the FKG inequalities, Commun. math. phys., 74, 119-128, (1980) · Zbl 0429.60096 [8] Newman, C.M., 1984. Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y.L. (Ed.), Inequalities in Statistics and Probability. IMS Lecture Notes—Monograph Series, Vol. 5, pp. 127-140. [9] Nummelin, E., Large deviations for funccionals of stationary processes, Probab. theory. rel. fields, 86, 387-401, (1990) · Zbl 0685.60029 [10] Roussas, G.G., Asymptotic normality of a smooth estimate of a random field distribution function under association, Statist. probab. lett., 24, 77-90, (1995) · Zbl 0830.62040 [11] Roussas, G.G., 1999. Positive and negative dependence with some statistical applications. In: Ghosh, S. (Ed.), Asymptotics, Nonparametrics and Time Series, Statist. Textbooks Monogr. 158, Dekker, New York, pp. 757-788. · Zbl 1069.62518 [12] Sadikova, S.M., Two-dimensional analogies of an inequality of Esseen with applications to the central limit theorem, Theory probab. appl., 11, 325-335, (1996)
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