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Self-normalized Cramér type moderate deviations for stationary sequences and applications. (English) Zbl 1445.60027
Summary: Let $$(X_i)_{i \geq 1}$$ be a stationary sequence. Denote $$m = \lfloor n^\alpha \rfloor, 0 < \alpha < 1,$$ and $$k = \lfloor n / m \rfloor$$, where $$\lfloor a \rfloor$$ stands for the integer part of $$a$$. Set $$S_j^\circ = \sum_{i = 1}^m X_{m (j - 1) + i}, 1 \leq j \leq k,$$ and $$(V_k^\circ)^2 = \sum_{j = 1}^k (S_j^\circ)^2$$. We prove a Cramér type moderate deviation expansion for $$\mathbb{P} (\sum_{j = 1}^k S_j^\circ / V_k^\circ \geq x)$$ as $$n \to \infty$$. Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
MSC:
 60F10 Large deviations 60G10 Stationary stochastic processes 60E15 Inequalities; stochastic orderings
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References:
 [1] Bercu, B.; Touati, A., Exponential inequalities for self-normalized martingales with applications, Ann. Appl. Probab., 18, 5, 1848-1869 (2008) · Zbl 1152.60309 [2] Bikelis, A., Estimates of the remainder in the central limit theorem, Litovsk. Mat. Sb, 6, 3, 323-346 (1966) [3] Caron, E.; Dede, S., Asymptotic distribution of the least squares estimators for linear models with dependent errors: regular designs, Math. Methods Statist., 27, 4, 268-293 (2018) · Zbl 1425.62086 [4] Chen, X.; Shao, Q. M.; Wu, W. B.; Xu, L., Self-normalized Cramér-type moderate deviations under dependence, Ann. Statist., 44, 4, 1593-1617 (2016) · Zbl 1359.62060 [5] Chung, K. L., The approximate distribution of student’s statistic, Ann. Math. Statist, 17, 4, 447-465 (1946) · Zbl 0063.00894 [6] Cramér, H., Sur un nouveau théorème-limite de la théorie des probabilités, Actualite’s Sci. Indust, 736, 5-23 (1938) · JFM 64.0529.01 [7] Csörgő, M.; Szyszkowicz, B.; Wang, Q., Donsker’s theorem for self-normalized partial sums processes, Ann. Probab., 31, 3, 1228-1240 (2003) · Zbl 1045.60020 [8] Cuny, C.; Merlevède, F., On martingale approximations and the quenched weak inviariance principle, Ann. Probab., 42, 2, 760-793 (2014) · Zbl 1354.60031 [9] Dedecker, J.; Merlevède, F.; Peligrad, M.; Utev, S., Moderate deviations for stationary sequences of bounded random variables, Ann. Inst. H. Poincaré Probab. Statist., 45, 2, 453-476 (2009) · Zbl 1172.60005 [10] Dembo, A.; Shao, Q. M., Large and moderate deviations for hotelling’s $$T^2$$-statistics, Electron. Comm. Proba., 11, 149-159 (2006) · Zbl 1112.60017 [11] Fan, X., Sharp large deviation results for sums of bounded from above random variables, Sci. China Math., 60, 12, 2465-2480 (2017) · Zbl 1382.60053 [12] Fan, X.; Grama, I.; Liu, Q., Cramér large deviation expansions for martingales under Bernstein’s condition, Stochastic Process. Appl, 123, 11, 3919-3942 (2013) · Zbl 1327.60069 [13] Fan, X.; Grama, I.; Liu, Q.; Shao, Q. M., Self-normalized Cramér type moderate deviations for martingales, Bernoulli, 25, 4A, 2793-2823 (2018) · Zbl 1428.62424 [14] Gao, F. Q., Moderate deviations for martingales and mixing random processes, Stochastic Process. Appl, 61, 263-275 (1996) · Zbl 0854.60028 [15] Giné, E.; Götze, F.; Mason, D. M., When is the student t-statistic asymptotically standard normal?, Ann. Probab., 25, 3, 1514-1531 (1997) · Zbl 0958.60023 [16] Grama, I., On moderate deviations for martingales, Ann. Probab., 25, 152-184 (1997) · Zbl 0881.60026 [17] Grama, I.; Haeusler, E., Large deviations for martingales via Cramér’s method, Stochastic Process. Appl., 85, 279-293 (2000) · Zbl 0997.60019 [18] Hannan, E. J., Central limit theorems for time series regression, Probab. Theory Related Fields, 26, 157-170 (1973) · Zbl 0246.62086 [19] Jing, B. Y.; Shao, Q. M.; Wang, Q., Self-normalized Cramér-type large deviations for independent random variables, Ann. Probab., 31, 4, 2167-2215 (2003) · Zbl 1051.60031 [20] Linnik, Y. V., On the probability of large deviations for the sums of independent variables, (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2 (1961), Univ of California Press.), 289-306 [21] Liu, W.; Shao, Q. M., A Cramér moderate deviation theorem for hotelling’s $$T^2$$-statistic with applications to global tests, Ann. Statist., 41, 1, 296-322 (2013) · Zbl 1347.62032 [22] Liu, W.; Shao, Q. M.; Wang, Q., Self-normalized Cramér type moderate deviations for the maximum of sums, Bernoulli, 19, 3, 1006-1027 (2013) · Zbl 1273.60032 [23] de la Peña, V. H., A general class of exponential inequalities for martingales and ratios, Ann. Probab., 27, 1, 537-564 (1999) · Zbl 0942.60004 [24] de la Peña, V. H.; Lai, T. L.; Shao, Q. M., (Self-Normalized Processes: Theory and Statistical Applications. Self-Normalized Processes: Theory and Statistical Applications, Springer Series in Probability and its Applications. (2009), Springer-Verlag: Springer-Verlag New York.) · Zbl 1165.62071 [25] Peligrad, M.; Utev, S.; Wu, W. B., A maximal $$L_p$$-inequality for stationary sequecens and its applications, Proc. Amer. Math. Soc., 135, 541-550 (2007) · Zbl 1107.60011 [26] Petrov, V. V., A generalization of Cramér’s limit theorem, Uspekhi Math. Nauk., 9, 195-202 (1954) [27] Q. M., Shao, Self-normalized large deviations, Ann. Probab., 25, 1, 285-328 (1997) · Zbl 0873.60017 [28] Račkauskas, A., Large deviations for martingales with some applications, Acta Appl. Math., 38, 109-129 (1995) · Zbl 0826.60021 [29] Račkauskas, A., Limit theorems for large deviations probabilites of certain quadratic forms, Lith. Math. J, 37, 402-415 (1997) · Zbl 0927.60043 [30] Rio, E., Moment inequalities for sums of dependent random variables under projective condition, J. Theor. Probab., 22, 146-163 (2009) · Zbl 1160.60312 [31] Saulis, L.; Statulevičius, V. A., Limit Theorems for Large Deviations (1978), Kluwer Academic Publishers · Zbl 0958.60026 [32] Shao, Q. M., A Cramér type large deviation result for student’s $$t -$$ statistic, J. Theor. Probab., 12, 2, 385-398 (1999) · Zbl 0927.60045 [33] Shao, Q. M., On necessary and sufficient conditions for the self-normalized central limit theorem, Sci. China Math., 61, 10, 1741-1748 (2018) · Zbl 1417.60018 [34] Shao, Q. M.; Wang, Q. Y., Self-normalized limit theorems: A survey, Probab. Surv., 10, 69-93 (2013) · Zbl 1286.60029 [35] Wu, W. B., Nonlinear system theorey: Another look at dependence, Proc. Natl. Acad. Sci. USA, 102, 14150-14154 (2005) · Zbl 1135.62075 [36] Wu, W. B.; Zhao, Z., Moderate deviations for stationary processes, Statist. Sinica, 18, 769-782 (2008) · Zbl 1152.62063
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