Strong Laws of Large Numbers for Intermediately Trimmed Sums of i.i.d. Random Variables with Infinite Mean

Abstract

Abstract. We show that for every sequence of non-negative i.i.d. random variables with infinite mean there exists a proper moderate trimming such that for the trimmed sum process a nontrivial strong law of large numbers holds. We provide an explicit procedure to find a moderate trimming sequence even if the underlying distribution function has a complicated structure, e.g. has no regularly varying tail distribution.

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

“…The results for general distribution functions are almost as strong as the i.i.d. trimming results in [KS17] and for the regular variation case we show that an intermediately trimmed strong law holds for the same trimming sequence (b n ) which in the i.i.d. case can be derived from [HM87].…”

“…This theorem is the equivalent to [KS17, Theorem B] for the setting of dynamical systems. In [KS17] we also give an example how to find a proper trimming function for a given distribution function F .…”

Trimmed sums for observables on the doubling map

“…From these results it is possible to establish an intermediately trimmed strong law. An intermediately trimmed strong law for more general distribution functions was also subject in [HM91] and [KS17b]. However, as can be seen from the above explanation, the behavior in this example differs fundamentally from the i.i.d.…”

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

“…For such distribution functions an intermediately trimmed strong law can be easily deduced from results by Haeusler and Mason, see [HM87], and a lower bound for the trimming sequence (b n ) can be derived from a result by Haeusler, see [Hae93]. In [HM91] and [KS17] intermediately trimmed strong laws for other distribution functions are given. The history for trimmed strong laws in the dynamical systems setting follows a similar line.…”