Optimal moment conditions for complete convergence for maximal normed weighted sums from arrays of rowwise independent random elements in Banach spaces

Abstract

This work establishes complete convergence for maximal normed weighted sums from arrays of rowwise independent random elements taking values in a real separable stable type p Banach space or in a real separable Rademacher type p Banach space under optimal moment conditions. An extension of a result in Hu et al. (Stochas Anal Appl 39:177–193, 2021) is obtained as a special case of the main theorem. To establish the main result, which is a Baum–Katz–Hsu–Robbins–Erdös-type theorem for maximal normed weighted sums, we prove a Rosenthal-type inequality for maximal normed partial sums of independent random elements taking values in Rademacher type p Banach spaces. Moreover, the conditions for complete convergence in the main result are shown to completely characterize stable type p Banach spaces when $1\le p< 2$. The sharpness of the results is illustrated by two examples.

A Appendix

In this section, we present some known results which are used in the previous sections. The first lemma is Theorem 1 in [24].

Lemma A.1

Let $\{X_i,1\le i\le n\}$ be a collection of n independent mean 0 random elements in a real separable Banach space ${\mathcal {X}}$. Then for all $q\ge 1$, there exists a constant $C_{q}$ depending only on q such that

$$\begin{aligned} \mathbb {E}\left( \left\| \sum _{i=1}^{n}X_i\right\| ^q\right) \le C_{q}\left( \left( \mathbb {E}\left\| \sum _{i=1}^n X_i\right\| \right) ^{q}+ \mathbb {E}\left( \max _{1\le i\le n}\Vert X_i\Vert ^{q}\right) \right) . \end{aligned}$$

The following simple result can be obtained by standard estimate (see Theorem 2.12.3 in [9] or Lemma 4 in Thành [26] for a more general version of condition (ii)).

Lemma A.2

Let $\alpha>0,\ q>p>0$ and let X be a real-valued random variable. Then the following three statements are equivalent.

• (i) $\mathbb {E}(|X|^p)<\infty$.

• (ii) $\sum _{n=1}^{\infty }n^{\alpha p-1}\mathbb {P}\left( |X|>n^{\alpha }\right) <\infty$.

• (iii) $\sum _{n=1}^{\infty }n^{\alpha (p-q)-1}\mathbb {E}\left( |X|^{q}\textbf{1}(|X|\le n^{\alpha })\right) <\infty$.

The next lemma is Lemma 2.3 in [20]. This lemma may be compared with Lemma 2.4 of [25] or Theorem V.9.1 of [28].

Lemma A.3

Let $1\le p<2$ and let ${\mathcal {X}}$ be a real separable Banach space. Suppose for every sequence $\{X_n,n\ge 1\}$ of independent and symmetric ${\mathcal {X}}$-valued random elements which is stochastically dominated by a real-valued random variable X with $\mathbb {E}(|X|^p)<\infty$ that

$$\begin{aligned} \lim _{n\rightarrow \infty }\dfrac{\sum _{i=1}^{n}X_i}{n^{1/p}}=0 \text { in probability}. \end{aligned}$$

Then ${\mathcal {X}}$ is of stable type p.