Limit theorems in the stadium billiard
Limit theorems in the stadium billiard
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DOI：
10.1007/s00220-005-1511-6
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发表时间：
2006-04-01
影响因子：
2.4
通讯作者：
Gouëzel, S
中科院分区：
文献类型：
--
作者：
Bálint, P;Gouëzel, S
我们证明了球场台球中“几乎所有”相关可观测值的Birkhoff和服从非标准极限定律。更准确地说，通常的中心极限定理对一个可观测值成立，当且仅当它沿一维不变集的积分消失，否则需要一个根号n log n的归一化。作为论证的两个关键步骤之一，我们得到了一个极限定理，该定理一般适用于具有指数返回时间统计的Young塔，这是一个抽象的结果，似乎适用于许多其他情况。
We prove that the Birkhoff sums for "almost every" relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a root n log n normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.