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

Abstract

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.

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

“…Our approach is to prove that (X, B, µ, χ) fulfilling Property F implies the existence of ǫ 0 ∈ (0, 1) such that X, B, σ, µ, H, · ǫ0 , χ fulfills Property D, see Lemma 3.13. Hence, all the statements given in [KS18] also hold for the tuple X, B, σ, µ, F , · ǫ0 , χ . For brevity we will not restate these results.…”

“…A similar Banach space was first considered by Blank, see [Bla97,Chapter 2.3], and Saussol, see [Sau00], in the context of multidimensional expanding maps. Furthermore, we prove some new limit theorems not given in [KS18] which we consider as particularly interesting for the application to observables on a subshift with a Gibbs-Markov measure, see Section 1.3. As a side result we obtain a limit theorem for sums of the truncated random variables.…”

“…It is the aim of the present paper to adapt these results to subshifts of finite type. It turns out that in contrary to the example of piecewise expanding interval maps it is not immediately clear how to apply the results from [KS18] to subshifts of finite type as the usually considered space of Lipschitz continuous functions does not fulfill all required properties of Property D. In particular, Property D requires ½ {χ>ℓ} ≤ K uniformly in ℓ which can not be fulfilled for an unbounded potential χ with respect to the Lipschitz norm, see Remark 1.2. Instead we consider the Banach space of quasi-Hölder continuous functions which is larger than the space of Lipschitz continuous functions but still obeys a spectral gap property.…”

A note on large deviations for unbounded observables

“…For more about these, see [KS17, Section 1.3]. These maps satisfy Property D introduced by Schindler [Sch15] and described also in [KS17], see the discussion preceding Proposition 6.1 in the Appendix.…”

“…For the upper bound we use recent results of Tanja Schindler [Sch15, Lemma 6.15], see Proposition 6.1 in the Appendix. Related estimates are given in [KS17].…”