1 Introduction
A finite sequence
\(\{X_{1}, \ldots, X_{n}\}\) of
\(\mathbb{R}^{d}\)-valued random vectors is said to be negatively associated (NA) if for any disjoint subsets
\(A, B \subset\{1,2,\dots, n\}\) and any real coordinatewise nondecreasing functions
f on
\(\mathbb{R}^{ \vert A \vert d}\) and
g on
\(\mathbb{R}^{ \vert B \vert d}\),
$$ \operatorname{Cov}\bigl(f(X_{i}, i\in A), g(X_{j}, j\in B)\bigr)\leq0, $$
(1.1)
whenever the covariance exists, where
\(\vert A \vert \) denotes the cardinality of a set
A. An infinite sequence
\(\{X_{n}, n\geq1\}\) of
\(\mathbb {R}^{d}\)-valued random vectors is NA if every finite subsequence is NA. This definition was introduced by Ko et al. [
1], and in the case
\(d=1\) the concept of negative association was introduced by Joag-Dev and Proschan [
2].
Let H be a real separable Hilbert space with the norm \(\Vert \cdot \Vert \) generated by an inner product \(\langle\cdot,\cdot\rangle\). Let \(\{ e_{j}, j\geq1\}\) be an orthonormal basis in H. For an H-valued random vector X, we denote \(X^{(j)}=\langle X, e_{j}\rangle\).
Ko et al. [
1] also extended the concept of NA for
\(\mathbb {R}^{d}\)-valued random vectors to random vectors with values in a real separable Hilbert space as follows: A sequence
\(\{X_{n}, n\geq1\}\) of random vectors taking values in a real separable Hilbert space
\((H, \langle\cdot,\cdot\rangle)\) is called NA if for some orthonormal basis
\(\{e_{k}, k\geq1\}\) in
H and for any
\(d\geq1\), the
d-dimensional sequence
\(\{(\langle X_{n}, e_{1}\rangle, \dots, \langle X_{n}, e_{d}\rangle), n\geq1\}\) is NA.
The definitions of NA random vectors in \(\mathbb{R}^{d}\) and in a Hilbert space can be applied to asymptotically almost negative association (AANA).
In the case
\(d=1\), the concept of AANA was introduced by Chandra and Ghosal [
3,
4]. Obviously, AANA random variables contain independent random variables (with
\(q(n)=0\) for
\(n\geq1\)) and NA random variables. Chandra and Ghosal [
3] pointed out that NA implies AANA, but AANA does not imply NA. Because NA has been applied to the reliability theory, multivariate statistical analysis, and percolation theory, the extension of the limit properties of NA random variables to AANA random variables is of interest in theory and applications.
Since the concept of AANA was introduced, various investigations have been established by many authors. For more detail, we can refer to Chandra and Ghosal [
3,
4], Ko et al. [
5], Yuan and An [
6,
7], Wang et al. [
8], Tang [
9], Shen and Wu [
10], and so forth.
The family of m-AANA sequence contains AANA (with \(m=1\)), NA, and independent sequences as particular cases.
2 Some lemmas
We start with the property of m-asymptotically almost negatively associated (m-AANA) random variables, which can be easily obtained from the definition of m-AANA random variables.
Nam et al. [
11] extended Lemma
2.2 to the case of
m-AANA sequence.
From Lemma
2.3 we obtain the Rosenthal-type inequality for
m-AANA random vectors with coefficients
\(\{q(n), n\geq1\}\) such that
\(\sum_{n=1}^{\infty}q^{2}(n)<\infty\) in a Hilbert space.
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