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Self-normalized limit theorems in probability and statistics. (English) Zbl 1269.60035

Lai, Tze Leung (ed.) et al., Asymptotic theory in probability and statistics with applications. Selected papers based on the presentations at the international conference, Hangzhou, China, Summer 2006. On the occasion of the 65th birthday of Professor Zhengyan Lin. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-169-8/hbk). Advanced Lectures in Mathematics (ALM) 2, 3-42 (2008).
Summary: The normalizing constants in classical limit theorems are usually sequences of real numbers. Moment conditions or other related assumptions are necessary and sufficient for many classical limit theorems. However, the situation becomes very different when the normalizing constants are sequences of random variables. A self-normalized large deviation shows that no any moment condition is needed for such large deviation type results. A self-normalized law of the iterated logarithm remains valid for all distributions in the domain of attraction of a normal or stable law. This reveals that self-normalization preserves essential properties much better than deterministic normalization does. In this chapter we review some important developments on self-normalized limit theorems in the last decade, especially on self-normalized large deviations, self-normalized saddle point approximations, self-normalized moderate deviations, self-normalized Cramér type large deviations for independent random variables, self-normalized law of the iterated logarithm and increments, self-normalized large and moderate deviations in \(\mathbb{R}^d\), large and moderate deviations of Hotelling’s \(T^2\) statistic, large and moderate deviations for self-normalized empirical processes; limiting distributions of self-normalized sums, weak invariance principle for self-normalized partial sum processes, Darling-Erdős type theorem, asymptotic distributions of non-central self-normalized sums, the pseudo-maximization approach for self-normalized stochastic processes.
For the entire collection see [Zbl 1144.60002].

MSC:

60F15 Strong limit theorems
60F05 Central limit and other weak theorems
60F10 Large deviations
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