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Asymptotic behavior of small ball probabilities. (English) Zbl 0994.60017

Grigelionis, B. (ed.) et al., Probability theory and mathematical statistics. Proceedings of the 7th international Vilnius conference, Vilnius, Lithuania, August, 12-18, 1998. Vilnius: TEV. 453-468 (1999).
This article is a survey of recent research about asymptotic behavior of small ball probabilities. The typical problem of the small ball probabilities is to investigate the asymptotic behavior of \(P{ ||X||<r},\) as \(r\) goes to zero, for a random vector \(X\) taking values in a normed space. The subject is known for long time to be hard but there was substantial progress during the last years. The author surveys the recent results with special focus on the fruitful connection between Gaussian small ball probabilities and analytical problems of approximation theory. The author presents several interesting open problems, too.
For the entire collection see [Zbl 0968.00043].

MSC:

60F05 Central limit and other weak theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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