Self-normalized large deviations
Self-normalized large deviations
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DOI：
10.1214/aop/1024404289
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发表时间：
1997
影响因子：
2.3
通讯作者：
Q. Shao
中科院分区：
文献类型：
--
作者：
Q. Shao
设{X，Xn，n≥1}是独立同分布的随机变量序列。经典的Cramer-Chernoff大偏差表示：LIMn→∞n-1lnP((ΣIn=1Xi)/n≥x)=lnp(X)当且仅当X的矩母函数在零的右邻域内有限。本文以n(p-1/pv n，p=n(p-1)/p(Σni=1|xi|1/p(p>；1))为归一化常数，建立了不带任何矩条件的自归一化大偏差。在正态或稳定律的吸引域中，对任意X，也找到了自归一化中偏差，即P(S n/Vn，p≥xn)的渐近概率.作为结果，得到了Griffin和Kuelbs的重对数律的自归一化的精确常数。文中还讨论了t统计量的渐近概率以及在Erdos-Renyi-Sepp大数定律中的应用。
Let {X, X n , n ≥ 1} be a sequence of independent and identically distributed random variables. The classical Cramer-Chernoff large deviation states that lim n→ ∞ n -1 ln P((Σ i n =1 X i )/n ≥ x) = ln p(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 / pV n,p = n (p-1)/p (Σ n i=1 |X i |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 ≥ 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 Erdos-Renyi-Shepp law of large numbers are also discussed.