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 Probability Surveys > Vol. 10 (2013) open journal systems 

Self-normalized limit theorems: A survey

Qi-Man Shao, Chinese University of Hong Kong
Qiying Wang, University of Sydney

Let \(X_1, X_2, \ldots,\) be independent random variables with \(EX_i=0\) and write \(S_n=\sum_{i=1}^nX_i\) and \(V_n^2=\sum_{i=1}^nX_i^2\). This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum \(S_n/V_n\). Other self-normalized limit theorems are also briefly discussed.

AMS 2000 subject classifications: Primary 60F05, 60F17; secondary 62E20.

Keywords: Self-normalized sum, Student t statistic, central limit theorem, invariance principle, convergence rate, absolute error, relative error, Cramér moderate deviation, large deviation, saddle-point approximation, laws of the iterated logarithm, Darling-Erdös theorem, Hotelling’s T2 statistic.

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Shao, Qi-Man, Wang, Qiying, Self-normalized limit theorems: A survey, Probability Surveys, 10, (2013), 69-93 (electronic). DOI: 10.1214/13-PS216.


[1]    Aleshkyavichene, A. K. (1979). Probabilities of large deviations for the maximum of sums of independent random variables. Theory Probab. Appl., 24, 16–33. MR0522234

[2]    Aleshkyavichene, A. K. (1979). Probabilities of large deviations for the maximum of sums of independent random variables. II. Theory Probab. Appl., 24, 318–331. MR0532445

[3]    Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. Third edition. Wiley-Interscience. MR1990662

[4]    Arvesen, J. N. (1969). Jackknifing U-statistics. Ann. Math. Statist., 40, 2076–2100. MR0264805

[5]    Balan, R. and Kulik, R. (2009). Weak invariance principle for mixing sequences in the domain of attraction of normal law. Studia Sci. Math. Hungar., 46, 329–343. MR2657020

[6]    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. MR1400598

[7]    Bentkus, V. and Götze, F. (1996). The Berry-Esséen bound for Student’s statistic. Ann. Probab., 24, 491–503. MR1387647

[8]    Bentkus, V., Jing, B.-Y., Shao, Q.-M. and Zhou, W. (2007). Limit distributions of non-central t-statistic and their applications to the power of t-test under non-normality. Bernoulli, 13, 346–364. MR2331255

[9]    Bloznelis, M. and Putter, H. (1998). One term Edgeworth expansion for Student’s t statistic. Probability Theory and Mathematical Statistics: Proceedings of the Seventh Vilnius Conference (B. Grigelionis et al., eds.), Vilnius, Utrecht: VSP/TEV, 1999, 81–98.

[10]    Bloznelis, M. and Putter, H. (2002). Second order and Bootstrap approximation to Student’s statistic. Theory Probab. Appl., 47, 374–381. MR2003205

[11]    Chistyakov, G. P. and Götze, F. (2003). Moderate deviations for Student’s statistic. Theory Probab. Appl., 47, 415–428. MR1975426

[12]    Chistyakov, G. P. and Götze, F. (2004a). On bounds for moderate deviations for Student’s statistic. Theory Probab. Appl., 48, 528–535. MR2141355

[13]    Chistyakov, G. P. and Götze, F. (2004b). Limit distributions of Studentized means. Ann. Probab., 32, no. 1A, 28–77. MR2040775

[14]    Choi, Y.-K. and Moon, H.-J. (2010). Self-normalized weak limit theorems for a ϕ-mixing sequence. Bull. Korean Math. Soc., 47, 1139–1153. MR2560606

[15]    Csörgʺo  , M. and Horváth, L. (1988). Asymptotic representations of self-normalized sums. Probability and Mathematical Statistics, 9, no. 1, 15–24. MR0945676

[16]    Csörgʺo    , M. and Hu, Z. (2013). A strong approximation of self-normalized sums. Sci. China Math., 56, 149–160. MR3016589

[17]    Csörgʺo   , M., Hu, Z. and Mei, H. (2013). Strassen-type law of the iterated logarithm for self-normalized sums. J. Theoret. Probab., 26, 311–328. MR3055805

[18]    Csörgʺo      , M., Szyszkowicz, B. and Wang, Q. (2003a). Darling-Erd˝o   s theorems for self-normalized sums. Ann. Probab., 31, 676–692. MR1964945

[19]    Csörgʺo  , M., Szyszkowicz, B. and Wang, Q. (2003b). Donsker’s theorem for self-normalized partial sums processes. Ann. Probab., 31, 1228–1240. MR1988470

[20]    Csörgʺo    , M., Szyszkowicz, B. and Wang, Q. (2004). On weighted approxiamtions and strong limit theorems for self-normalized partial sums processes. In Asymptotics Methods in Stochastics: Festschrift for M. Csörgö (L. Horváth and B. Szyszkowicz, eds.), Fields institute communications (Vol. 44), 489–521. MR2108953

[21]    Csörgʺo , M., Szyszkowicz, B. and Wang, Q. (2008). On weighted approximations in D[0;1] with applications to self-normalized partial sum processes. Acta Math. Hungar., 121, 333–357. MR2461439

[22]    de la Peña, V., Lai, T. L. and Shao, Q. M. (2009). Self-normalized Processes: Theory and Statistical Applications. Springer, New York. MR2488094

[23]    Dembo, A. and Shao, Q.-M. (1998). Self-normalized moderate deviations and LILs. Stochastic Process. Appl., 75, 51–65. MR1629018

[24]    Dembo, A. and Shao, Q.-M. (2006). Large and moderate deviations for Hotelling’s t2-statistic. Elect. Comm. in Probab., 11, 149–159. MR2240708

[25]    Efron, B. (1969). Student’s t-test under symmetry conditions. J. Amer. Statist. Assoc., 1278–1302. MR0251826

[26]    Egorov, V. A. (1996). On the asymptotic behavior of self-normalized sums of random variables, Teor. Veroyatnost. i Primenen., 41, 643–650. MR1450081

[27]    Egorov, V. A. (2002). An estimate for the tail of the distribution of normalized and self-normalized sums. J. Math. Sci. (N. Y.), 127, 1717–1722. MR1976748

[28]    Einmahl, U. (1989). The Darling-Erd˝o  s theorem for sums of i.i.d. random variables, Probab. Th. Rel. Fields, 82, 241–257. MR0998933

[29]    Giné, E., Götze, F. and Mason, D. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Probab., 25, 1514–1531. MR1457629

[30]    Giné, E. and Mason, D. (1998). On the LIL for self-normalized sums of iid random variables. J. Theoret. Probab., 18, 351–370. MR1622575

[31]    Griffin, P. S. and Kuelbs, J. D. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab., 17, 1571–1601. MR1048947

[32]    Griffin, P. S. and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums. Math. Proc. Cambridge Philos. Soc., 109, 597–610. MR1094756

[33]    Hall, P. (1987). Edgeworth expansion for Student’s t statistic under minimal moment conditions. Ann. Probab., 15, 920–931. MR0893906

[34]    Hall, P. (1988). On the effect of random norming on the rate of convergence in the central limit theorem. Ann. Probab., 16, 1265–1280. MR0942767

[35]    Hall, P. and Wang, Q. (2004). Exact convergence rate and leading term in central limit theorem for student’s t statistic. Ann. Probab., 32, 1419–1437. MR2060303

[36]    Hu, Z., Shao, Q.-M. and Wang, Q. (2009). Cramér type large deviations for the maximum of self-normalized sums. Electronic Jour. of Probab., 14, 1181–1197. MR2511281

[37]    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. MR2016616

[38]    Jing, B.-Y., Shao, Q.-M. and Zhou, W. (2004). Saddlepoint approximation for Student’s t-statistic with no moment conditions. Ann. Statist., 6, 2679–2711. MR2153999

[39]    Jing, B.-Y., Shao, Q.-M. and Zhou, W. (2008). Towards a universal self-normalized moderate deviation. Trans. Amec. Math. Soci., 360, 4263–4285. MR2395172

[40]    Kulik, R. (2006). Limit theorems for self-normalized linear processes. Statist. Probab. Lett., 76, 1947–1953. MR2329238

[41]    Lai, T. L., Shao, Q.-M., and Wang, Q. (2011). Cramér type moderate deviations for Studentized U-statistics. ESAIM – Probability and Statistics, 15, 168–179. MR2870510

[42]    Liu, W. and Shao, Q.-M. (2012). Cramér moderate deviation for Hotelling’s T2 statistic with applications to global tests. Preprint.

[43]    Liu, W., Shao, Q.-M. and Wang, Q. (2012). Self-normalized Cramér type moderate deviations for the maximum of sums. Bernoulli, forthcoming. MR3079304

[44]    Logan, B. F., Mallows, C. L., Rice, S. O. and Sheep, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab., 1, 788–809. MR0362449

[45]    Maller, R. A. (1981). A theorem on products of random variables with application to regression. Australian Journal of Statistics, 23, 177–185. MR0636133

[46]    Mason, D. M. (2005). The asymptotic distribution of self-normalized triangular arrys. Journal of Theoret. Probab., 18, 853–870. MR2289935

[47]    Novak, S. Y. (2004). On self-normalized sums of random variables and the Student’s statistic. Theory Probab. Appl., 49, 336–344. MR2144306

[48]    O’Brien, G. L. (1980). A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Probab., 17, 539–545. MR0568964

[49]    De la Pena, V. H. and Giné, E. (1999). Decoupling: From Dependence to Independence. Springer, New York. MR1666908

[50]    De la Pena, V. H., Klass, M. J. and Lai, T. L. (2000). Moment bounds for self-normalized martingales. In High Dimensional Probability II (E. Giné, D. M. Mason and J. A. Wellner, eds.), Birkhauser, Boston, 1–11. MR1857343

[51]    De la Pena, V. H., Klass, M. J. and Lai, T. L. (2004). Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws. Ann. Probab., 32, 1902–1933. MR2073181

[52]    De la Pena, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-normalized Processes: Limit Theorey and Statistical Applications. Springer, New York. MR2488094

[53]    Prohorov, Yu. V. (1956). Covergence of random processes and limit theorems in probability theory. Theor. Probab. Appl., 1, 157–214. MR0084896

[54]    Robinson, J. and Wang, Q. (2005). On the self-normalized Cramér-type large deviation. Journal of Theoret. Probab., 18, 891–909. MR2300002

[55]    Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab., 25, 285–328. MR1428510

[56]    Shao, Q.-M. (1998). Recent development in self-normalized limit theorems. In: Asymptotic Methods in Probability and Statistics (B. Szyszkowicz, ed.), Elsevier Science, 467–480. MR1661499

[57]    Shao, Q.-M. (1999). A Cramér type large deviation result for Student’s t-statistic. J. Theoret. Probab., 12, 385–398. MR1684750

[58]    Shao, Q.-M. (2004). Recent progress on self-normalized limit theorems. In: Probability, Finance And Insurance, World Sci. Publ., River Edge, NJ, 50–68. MR2189198

[59]    Shao, Q.-M. (2005). An explicit Berry-Esseen bound for Student’s t-statistic via Stein’s method. In Stein’s Method and Applications, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 143–155. MR2205333

[60]    Shao, Q.-M. (2010). Stein’s method, self-normalized limit theory and applications. In Proceedings of the International Congress of Mathematicians, Hyderabad, India, 1–25.

[61]    Shao, Q. M. and Zhou, W. (2012). Cramér type moderate deviation theorems for Studentized non-linear statistics. Preprint.

[62]    Shao, Q. M., Zhang, K. and Zhou, W. (2012). Berry-Esseen bounds for Studentized non-linear statistics via Stein’s method. Preprint.

[63]    Slavov, V. V. (1985). On the Berry-Esseen bound for Student’s statistic. In Stability Problems for Stochastic Models (V. V. Kalashnikov and V. M. Zolotarev, eds.), Lecture Notes in Math., 1155, Springer, Berlin, 355–390. MR0825335

[64]    Sepanski, S. (1994). Asymptotics for Multivariate t-statistic and Hotelling’s T2-statistic under infinite second moments via bootstrapping. J. Multivariate Anal., 49, 41–54. MR1275042

[65]     Wang, Q. (2004). On Darling Erdös type theorems for self-nornmalized sums, In Asymptotic Methods in Stochastics: Festschrift for Miklos Csorgo (Lajos Horvath and Barbara Szyszkowicz, eds.), International Conference on Asymptotic Methods in Stochastics. Fields Institute Communications, American Mathematical Society, 523–530. MR2108954

[66]    Wang, Q. (2005). Limit theorems for self-normalized large deviation. Electronic Journal of Probab., 10, 1260–1285. MR2176384

[67]    Wang, Q. (2011). Refined self-normalized large deviations for independent random variables. Journal of Theor. Probab., 24, 307–329. MR2795041

[68]    Wang, Q. and Hall, P. (2009). Relative errors in central limit theorem for Student’s t statistic, with applications. Statistica Sinica, 19, 343–354. MR2487894

[69]    Wang, Q. and Jing, B.-Y. (1999). An exponential nonuniform Berry-Esseen bound for self-normalized sums. Ann. Probab., 27, 2068–2088. MR1742902

[70]    Wang, Q., Jing, B.-Y. and Zhao, L. (2000). The Berry-Esseen bound for Studentized statistics. Ann. Probab., 28, 511–535. MR1756015

[71]    Zhou, W. and Jing, B.-Y. (2006). Tail probability approximations for Student’s t statistics. Probab. Theory Relat. Fields, 136, 541–559. MR2257135

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