• Bernoulli
  • Volume 25, Number 4A (2019), 2793-2823.

Self-normalized Cramér type moderate deviations for martingales

Xiequan Fan, Ion Grama, Quansheng Liu, and Qi-Man Shao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $(X_{i},\mathcal{F}_{i})_{i\geq 1}$ be a sequence of martingale differences. Set $S_{n}=\sum_{i=1}^{n}X_{i}$ and $[S]_{n}=\sum_{i=1}^{n}X_{i}^{2}$. We prove a Cramér type moderate deviation expansion for $\mathbf{P}(S_{n}/\sqrt{[S]_{n}}\geq x)$ as $n\to +\infty $. Our results partly extend the earlier work of Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) for independent random variables.

Article information

Bernoulli, Volume 25, Number 4A (2019), 2793-2823.

Received: February 2018
Revised: June 2018
First available in Project Euclid: 13 September 2019

Permanent link to this document

Digital Object Identifier

Cramér’s moderate deviations martingales self-normalized sequences


Fan, Xiequan; Grama, Ion; Liu, Quansheng; Shao, Qi-Man. Self-normalized Cramér type moderate deviations for martingales. Bernoulli 25 (2019), no. 4A, 2793--2823. doi:10.3150/18-BEJ1071.

Export citation


  • [1] Bentkus, V., Bloznelis, M. and Götze, F. (1996). A Berry–Esséen bound for Student’s statistic in the non-i.i.d. case. J. Theoret. Probab. 9 765–796.
  • [2] Bentkus, V. and Götze, F. (1996). The Berry–Esseen bound for Student’s statistic. Ann. Probab. 24 491–503.
  • [3] Bercu, B., Delyon, B. and Rio, E. (2015). Concentration Inequalities for Sums and Martingales. SpringerBriefs in Mathematics. Cham: Springer.
  • [4] Bercu, B. and Touati, A. (2008). Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 1848–1869.
  • [5] Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 672–688.
  • [6] Chen, S., Wang, Z., Xu, W. and Miao, Y. (2014). Exponential inequalities for self-normalized martingales. J. Inequal. Appl. 2014 289.
  • [7] Chung, K.-L. (1946). The approximate distribution of Student’s statistic. Ann. Math. Stat. 17 447–465.
  • [8] Delyon, B. (2009). Exponential inequalities for sums of weakly dependent variables. Electron. J. Probab. 14 752–779.
  • [9] de la Peña, V.H. (1999). A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27 537–564.
  • [10] de la Peña, V.H., Lai, T.L. and Shao, Q.-M. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Probability and Its Applications (New York). Berlin: Springer.
  • [11] de la Peña, V.H. and Pang, G. (2009). Exponential inequalities for self-normalized processes with applications. Electron. Commun. Probab. 14 372–381.
  • [12] Esscher, F. (1924). On a method of determining correlation from the ranks of the variates. Scand. Actuar. J. 1 201–219.
  • [13] Fan, X., Grama, I., Liu, Q. and Shao, Q.-M. (2018). Supplement to “Self-normalized Cramér type moderate deviations for martingales.” DOI:10.3150/18-BEJ1071SUPP.
  • [14] Giné, E., Götze, F. and Mason, D.M. (1997). When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 1514–1531.
  • [15] Grama, I. and Haeusler, E. (2000). Large deviations for martingales via Cramér’s method. Stochastic Process. Appl. 85 279–293.
  • [16] Haeusler, E. (1988). On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab. 16 275–299.
  • [17] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application: Probability and Mathematical Statistics. New York: Academic Press [Harcourt Brace Jovanovich, Publishers].
  • [18] Hu, Z., Shao, Q.-M. and Wang, Q. (2009). Cramér type moderate deviations for the maximum of self-normalized sums. Electron. J. Probab. 14 1181–1197.
  • [19] Itô, K. and McKean, H.P. Jr. (1996). Difussion Processes and Their Sample Paths. Berlin: Springer.
  • [20] Jing, B., Liang, H. and Zhou, W. (2012). Self-normalized moderate deviations for independent random variables. Sci. China Math. 55 2297–2315.
  • [21] Jing, B.-Y., Shao, Q.-M. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167–2215.
  • [22] Linnik, Ju.V. (1961). On the probability of large deviations for the sums of independent variables. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II 289–306. Berkeley, CA: Univ. California Press.
  • [23] Liu, W., Shao, Q.-M. and Wang, Q. (2013). Self-normalized Cramér type moderate deviations for the maximum of sums. Bernoulli 19 1006–1027.
  • [24] Novak, S.Y. (2011). Extreme Value Methods with Applications to Finance. Monographs on Statistics and Applied Probability 122. Boca Raton, FL: CRC Press.
  • [25] Petrov, V.V. (1975). Sums of Independent Random Variables. New York: Springer. Translated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.
  • [26] Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285–328.
  • [27] Shao, Q.-M. (1999). A Cramér type large deviation result for Student’s $t$-statistic. J. Theoret. Probab. 12 385–398.
  • [28] Shao, Q.-M. and Wang, Q. (2013). Self-normalized limit theorems: A survey. Probab. Surv. 10 69–93.
  • [29] Shao, Q.-M. and Zhou, W.-X. (2016). Cramér type moderate deviation theorems for self-normalized processes. Bernoulli 22 2029–2079.
  • [30] Slavova, V.V. (1985). On the Berry–Esseen bound for Student’s statistic. In Stability Problems for Stochastic Models (Uzhgorod, 1984). Lecture Notes in Math. 1155 355–390. Berlin: Springer.
  • [31] Talagrand, M. (1995). The missing factor in Hoeffding’s inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 31 689–702.
  • [32] von Bahr, B. and Esseen, C.-G. (1965). Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$. Ann. Math. Stat. 36 299–303.

Supplemental materials

  • Supplement to “Self-normalized Cramér type moderate deviations for martingales”. The supplement gives the detailed proofs of Propositions 3.1 and 3.2.