期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2017
卷号：XIV
页码：245-247
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：This note corrects a condition in Theorem 3.5 in our paper Fissler andTh¨ale (2016).It has been pointed out to us that the assertions of the equivalence stated inTheorem 3.5(b) of our paper Fissler and Th¨ale (2016) cannot be satisfied by asequence of kernels fn ∈ L2s(μqn) in the case q = 4. Indeed, the sign conditionfn ≤ 0 implies that E[Iq(fn)3] ≤ 0 in view of equation (2.8) in Fissler and Th¨ale(2016). On the other hand, if Iq(fn) converges in distribution to Y ∼ , as n → ∞,the uniform integrability of {Iq(fn)4 : n ≥ 1} implies thatlimn→∞E[Iq(fn)3] = E[Y 3] = 8ν > 0,which is a contradiction. In line with the equivalence, neither assertion (ii) nor (iii)can be satisfied if fn ≤ 0. E.g. c2q kfnk2 ≤ kfne⋆q/2q/2fn − cq fnk2 → 0 for fn ≤ 0,but at the same time q!kfnk2 → 2ν > 0. This contradiction also affects the resultsbased on Theorem 3.5(b), namely Corollary 3.8(b), Theorem 4.6(b), and Theorem4.10(b).
