具有正则变化尾的中等裁剪Birkhoff和的平均收敛性
Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails
摘要：
在具有谱隙性质的测度论动力系统上，我们考虑了具有规则变化尾的不可积观测量，并满足一个温和的混合条件。我们表明，这些可观的赋范修剪和过程，然后收敛于平均。这一结果对于i.i.d.的特殊情况也是新的。随机变量，并对比了平均收敛可能失败的一般情况，即使强大数律成立。为了说明所需的混合条件，我们给出了一个明确的例子，一个动力系统满足频谱间隙属性和观察定期变化的尾巴，但没有假设的混合条件，使平均收敛失败。
英文摘要：
On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails and fulfilling a mild mixing condition. We show that the normed trimmed sum process of these observables then converges in mean. This result is new also for the special case of i.i.d. random variables and contrasts the general case where mean convergence might fail even though a strong law of large numbers holds. To illuminate the required mixing condition we give an explicit example of a dynamical system fulfilling a spectral gap property and an observable with regularly varying tails but without the assumed mixing condition such that mean convergence fails.
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