Convergent subseries of divergent series

Abstract

Let $$\mathscr {X}$$ X be the set of positive real sequences $$x=(x_n)$$ x = ( x n ) such that the series $$\sum _n x_n$$ ∑ n x n is divergent. For each $$x \in \mathscr {X}$$ x ∈ X , let $$\mathcal {I}_x$$ I x be the collection of all $$A\subseteq \mathbf {N}$$ A ⊆ N such that the subseries $$\sum _{n \in A}x_n$$ ∑ n ∈ A x n is convergent. Moreover, let $$\mathscr {A}$$ A be the set of sequences $$x \in \mathscr {X}$$ x ∈ X such that $$\lim _n x_n=0$$ lim n x n = 0 and $$\mathcal {I}_x\ne \mathcal {I}_y$$ I x ≠ I y for all sequences $$y=(y_n) \in \mathscr {X}$$ y = ( y n ) ∈ X with $$\liminf _n y_{n+1}/y_n>0$$ lim inf n y n + 1 / y n > 0 . We show that $$\mathscr {A}$$ A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals $$\mathcal {I}_x$$ I x . This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.

The maximum domain of attraction of multivariate extreme value distributions is small

“…Results in the same spirit, but completely different contexts, appeared, e.g., in [2,3,14,15]. In Section 2 we collect some preliminary results, while in Section 3 we provide the proofs of Theorem 1.2 and Corollary 1.3.…”