Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type

Strong laws of large number for intermediately trimmed Birkhoff sums of observables with infinite mean

We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necess…

Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

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 w…

截断强律与指数混合系统中的分布极限

Trimmed strong laws and distributional limits for exponentially mixing systems

本文研究了指数混合系统中截断遍历和的强律和分布极限，证明了α≥1时的截断大数定律，并在α>1/2时展示了非标准分布和正态分布的收敛性。