Setting and definitions
-----------------------


Let $X = \{X(t), t \in T \}$, $T \subset \mathbb{Z}$, be an infinite-variance associated stochastic process, i.e.
$$
\text{Cov}(f(X(I)), g(X(J))) \geq 0
$$
for all finite disjoint subsets $I, J \subset T$ and bounded, coordinate-wise increasing Borel functions $f: \mathbb{R}^{\vert I \vert} \rightarrow \mathbb{R}$, $g: \mathbb{R}^{\vert J \vert} \rightarrow \mathbb{R}$.


A stochastic process $Y$ is called (BL, $\theta$)-dependent if if there exists
a non-increasing sequence $\theta = (\theta\_r)\_{r \in \mathbb{Z}}$ with $\theta\_r \rightarrow 0$ as $r \rightarrow \infty$ and
$$
 \Big\vert
 \text{Cov}
 \Big( 
 f\big(Y(I)\big), g\big(Y(J)\big) 
 \Big)
 \Big\vert 
 \leq 
 \text{Lip}(f)\text{Lip}(g)(\vert I \vert \wedge \vert J \vert)
 \theta\_r
$$


for any bounded Lipschitz-continuous functions $f: \mathbb{R}^{\vert I \vert} \rightarrow \mathbb{R}$, $g: \mathbb{R}^{\vert J \vert} \rightarrow \mathbb{R}$ and finite disjoint subsets $I, J \subset T$ such that $\text{dist}(I, J) := \min\{ \vert i - j \vert : i \in I, j \in J \} = r$.


Question
--------


Is $X$ (BL, $\theta$)-dependent? The finite-variance case can be proven as seen below. But the proof relies on covariances of the process and I don't know how to generalize the main inequality that was used in the proof.


Proof for finite-variance case
------------------------------


If $X$ had a finite-variance, then Theorem 5.3. in [Bulinski & Shashkin (2007)](https://www.worldscientific.com/worldscibooks/10.1142/6555#t=aboutBook) states that
$$
 \Big\vert
 \text{Cov}
 \Big( 
 f\big(X(I)\big), g\big(X(J)\big) 
 \Big)
 \Big\vert 
 \leq 
 \sum\_{i \in I} \sum\_{j \in J} \text{Lip}\_i(f)\text{Lip}\_j(g)
 \text{Cov}(X(i), X(j))
$$
for all any bounded Lipschitz-continuous functions $f: \mathbb{R}^{\vert I \vert} \rightarrow \mathbb{R}$, $g: \mathbb{R}^{\vert J \vert} \rightarrow \mathbb{R}$ and finite disjoint subsets $I, J \subset T$.
Hence, $X$ is (BL, $\theta$)-dependent with
$$
\theta\_r := \sup\_{i \in I} \sum\_{j \in \mathbb{Z} : \vert i - j \vert \geq r} \vert \text{Cov}(X(i), X(j)) \vert
$$
under the assumption that these quantities exist and tend to zero.



Infinite-variance associated processes are (BL, $\theta$)-dependent
The answer is no. E.g., Let $X(t)=Z$ for all $t$, where $Z$ is any random variable with infinite variance. Then the process $(X(t)\colon t\in\mathbb Z)$ is positively associated. On the other hand, for $f\_n(x):=\min(n,\max(-n,x))$ and each natural $j$ we have
$$Cov(f\_n(X\_0),f\_n(X\_r))\to Var\,Z=\infty$$
as $n\to\infty$, whereas
$$\text{Lip}(f\_n)\text{Lip}(f\_n)(|\{0\}|\wedge|\{r\}|)=1.$$


