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.

中文翻译：

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具有规律变化尾部的中间修剪 Birkhoff 和的平均收敛性

在具有光谱间隙特性的测量理论动力学系统上，我们考虑具有规律变化的尾部并满足温和混合条件的不可积的可观测量。我们表明，这些可观测值的规范修整求和过程随后在均值上收敛。这个结果对于 iid 随机变量的特殊情况也是新的，并且与即使强数定律成立但均值收敛可能会失败的一般情况形成对比。为了阐明所需的混合条件，我们给出了一个动态系统的明确示例，该系统满足光谱间隙特性和具有规则变化尾部的可观测值，但没有假设的混合条件，因此平均收敛失败。