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January 1997 Self-normalized large deviations
Qi-Man Shao
Ann. Probab. 25(1): 285-328 (January 1997). DOI: 10.1214/aop/1024404289


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.


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Qi-Man Shao. "Self-normalized large deviations." Ann. Probab. 25 (1) 285 - 328, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60017
MathSciNet: MR1428510
Digital Object Identifier: 10.1214/aop/1024404289

Primary: 60F10 , 60F15
Secondary: 60G50 , 62E20

Keywords: $t$-statistic , large deviation , Law of the iterated logarithm , limit distribution , Moderate deviation , self-normalized partial sums , the Erdös-Rényi-Shepp law of large numbers

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
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