## The Annals of Probability

### Self-normalized large deviations

Qi-Man Shao

#### Abstract

Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed random variables. The classical Cramér-Chernoff large deviation states that $\lim_{n\to\infty} n^{-1} \ln P((\sum_{i=1}^n X_i)/n \geq x) = \ln \rho (x)$ if and only if the moment generating function of $X$ is finite in a right neighborhood of zero. This paper uses $n^{(p-1)/p} V_{n,p} = n^{(p-1)/p}(\sum_{i=1}^n |X_i|^p)^{1/p} (p > 1)$ as the normalizing constant to establish a self-normalized large deviation without any moment conditions. A self-normalized moderate deviation, that is, the asymptotic probability of $P(S_n/V_{n,p} \geq x_n) for$x_n = o(n^{(p-1)/p})$, is also found for any$X$in the domain of attraction of a normal or stable law. As a consequence, a precise constant in the self-normalized law of the iterated logarithm of Griffin and Kuelbs is obtained. Applications to the limit distribution of self-normalized sums, the asymptotic probability of the$t\$-statistic as well as to the Erdös-Rényi-Shepp law of large numbers are also discussed.

#### Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 285-328.

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404289

Digital Object Identifier
doi:10.1214/aop/1024404289

Mathematical Reviews number (MathSciNet)
MR1428510

Zentralblatt MATH identifier
0873.60017

#### Citation

Shao, Qi-Man. Self-normalized large deviations. Ann. Probab. 25 (1997), no. 1, 285--328. doi:10.1214/aop/1024404289. https://projecteuclid.org/euclid.aop/1024404289

#### References

• Bahadur, R. R. (1971). Some limit theorems in statistics. Regional Conference Series in Applied Mathematics 4. SIAM, Philadelphia.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
• Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493-507.
• Chung, K. L. (1974). A Course in Probability Theory. Academic Press, New York.
• Cram´er, H. (1938). Sur un nouveaux th´eor´eme limite de la th´eorie des probabilit´es. Actualit´es Sci. Indust. 736 5-23. Hermann, Paris.
• Cs ¨org o, M. and Horv´ath, L. (1988). Asymptotic representations of self-normalized sums. Probab. Math. Statist. 9 15-24.
• Cs ¨org o, M. and R´ev´esz, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
• Cs ¨org o, M. and Shao, Q. M. (1994). A self-normalized Erd os-R´enyi type strong law of large numbers. Stochastic Process. Appl. 50 187-196.
• Cs ¨org o, M. and Steinebach, J. (1981). Improved Erd os-R´enyi and strong approximation laws for increments of partial sums. Ann. Probab. 9 988-996.
• Cs ¨org o, S. (1979). Erd os-R´enyi laws. Ann. Statist. 7 772-787.
• Darling, D. A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95-107.
• De Acosta, A. (1988). Large deviations for vector-valued functionals of a Markov chain: lower bounds. Ann. Probab. 16 925-960.
• Deheuvels, P., Devroye, L. and Lynch, J. (1986). Exact convergence rate in the limit theorems of Erd os-R´enyi and Shepp. Ann. Probab. 14 209-223.
• Dembo, A. and Zeitouni, O. (1992). Large Deviations Techniques and Applications. Jones and Bartlett, Boston.
• Donsker, M. D. and Varadhan, S. R. S. (1987). Large deviations for noninteracting particle systems. J. Statist. Physics 46 1195-1232.
• Efron, B. (1969). Student's t-test under symmetry conditions. J. Amer. Statist. Assoc. 64 1278- 1302.
• Erd os, P. and R´enyi, A. (1970). On a new law of large numbers. J. Analyse Math. 23 103-111.
• Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Wiley, New York.
• Griffin, P. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17 1571-1601.
• Hahn, M. G., Kuelbs, J. and Weiner, D. C. (1990). The asymptotic joint distribution of selfnormalized censored sums and sums of squares. Ann. Probab. 18 1284-1341.
• Hotelling, H. (1961). The behavior of some standard statistical tests under nonstandard conditions. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 319-360. Univ. California Press, Berkeley.
• Karamata, J. (1933). Sur un mode de croissance r´eguli ere, th´eor emes fondamentaux. Bull. Soc. Math. France 61 55-62.
• Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of selfnormalized sums. Ann. Probab. 1 788-809.
• Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10 287-298.
• Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
• Petrov, V. V. and Sirokova, I. V. (1973). The exponential rate of convergence in the law of large numbers. Vestnik Leningrad. Univ. No. 7 Mat. Mekh. Astronom. 2 155-157.
• Royden, H. L. (1968). Real Analysisi, 2nd ed. Macmillan, New York.
• Shepp, L. A. (1966). A limit theorem concerning moving averages. Ann. Math. Statist. 35 424-428.
• Steinebach, J. (1980). Large deviation probabilities and some related topics. Carleton Math. Lecture Notes 28. Carleton Univ.
• Strassen, V. (1966). A converse to the law of the iterated logarithm.Wahrsch. Verw. Gebiete 4 265-268.
• Stroock, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, Berlin.