The Annals of Probability

Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws

Victor H. de la Peña, Michael J. Klass, and Tze Leung Lai

Full-text: Open access


Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables Bt>0 and At, let Yt(λ)=exp{λAtλ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt0At/{Bt(log logBt)1/2} and variants thereof, when EYt(λ)1 or when Yt(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with At=Mt and $B_{t}=\sqrt {\langle M\rangle _{t}}$ , and sums of conditionally symmetric variables di with At=i=1tdi and $B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}$ . A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in Rm, m1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving i=1tdi and i=1tdi2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.

Article information

Ann. Probab., Volume 32, Number 3 (2004), 1902-1933.

First available in Project Euclid: 14 July 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Martingales self-normalized inequalities iterated logarithm


de la Peña, Victor H.; Klass, Michael J.; Leung Lai, Tze. Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws. Ann. Probab. 32 (2004), no. 3, 1902--1933. doi:10.1214/009117904000000397.

Export citation


  • Bañuelos, R. and Moore, C. N. (1999). Probabilistic Behavior of Harmonic Functions. Birkhäuser, Boston.
  • Barlow, M. T., Jacka, S. D. and Yor, M. (1986). Inequalities for a pair of processes stopped at a random time. Proc. London Math. Soc. (3) 53 142--172.
  • Bentkus, V. and Götze, F. (1996). The Berry--Esseen bound for Student's statistic. Ann. Probab. 24 491--503.
  • Breiman, L. (1968). A delicate law of iterated logarithm for non-decreasing stable processes. Ann. Math. Statist. 39 1818--1824.
  • Caballero, M. E., Fernández, B. and Nualart, D. (1998). Estimation of densities and applications. J. Theoret. Probab. 11 831--851.
  • De La Peña, V. H. (1999). A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27 537--564.
  • De La Peña, V. H. and Giné, E. (1999). Decoupling: From Dependence to Independence. Springer, New York.
  • De La Peña, V. H., Klass, M. J. and Lai, T. L. (2000). Moment bounds for self-normalized processes. In High Dimensional Probability II (E. Giné, D. M. Mason and J. A. Wellner, eds.) 3--12. Birkhäuser, Boston.
  • Durrett, R. (1996). Probability: Theory and Applications, 2nd ed. Duxbury, Belmont, CA.
  • Einmahl, U. and Mason, D. (1989). Some results on the almost sure behavior of martingales. In Limit Theorems in Probability and Statistics (I. Berkes, E. Csáki and P. Révész, eds.) 185--195. North-Holland, Amsterdam.
  • Egorov, V. (1998). On the growth rate of moments of random sums. Preprint.
  • Freedman, D. (1973). Another note on the Borel--Cantelli lemma and the strong law, with the Poisson approximation as a by-product. Ann. Probab. 1 910--925.
  • Giné, E., Goetze, F. and Mason, D. (1997). When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 1514--1531.
  • Giné, E. and Mason, D. M. (1998). On the LIL for self-normalized sums of i.i.d. random variables. J. Theoret. Probab. 11 351--370.
  • Griffin, P. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17 1571--1601.
  • Griffin, P. and Kuelbs, J. (1991). Some extensions of the LIL via self-normalizations. Ann. Probab. 19 380--395.
  • Hitczenko, P. (1990). Upper bounds for the $L_p$-norms of martingales. Probab. Theory Related Fields 86 225--238.
  • Jing, B., Shao, Q. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167--2215.
  • Kikuchi, M. (1991). Improved ratio inequalities for martingales. Studia Math. 99 109--113.
  • Kubilius, K. and Mémin, J. (1994). Inégalité exponentielle pour les martingales locales. C. R. Acad. Sci. Paris Ser. I 319 733--738.
  • Lai, T. L. (1976). Boundary crossing probabilities for sample sums and confidence sequences. Ann. Probab. 4 299--312.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, New York.
  • Robbins, H. and Siegmund, D. (1970). Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Statist. 41 1410--1429.
  • Shao, Q. (1997). Self-normalized large deviations. Ann. Probab. 25 285--328.
  • Shao, Q. (2000). A comparison theorem on maximal inequalities between negatively associated and independent random variables. J. Theoret. Probab. 13 343--356.
  • Stout, W. F. (1970). A martingale analogue of Kolmogorov's law of iterated logarithm. Z. Wahrsch. Verw. Gebiete 15 279--290.
  • Stout, W. F. (1973). Maximal inequalities and the law of iterated logarithm. Ann. Probab. 1 322--328.
  • Wang, G. (1989). Some sharp inequalities for conditionally symmetric martingales. Ph.D. dissertation, Univ. Illinois at Urbana-Champaign.