We have a collection of random intervals $\{I_{k}:=(X_{k},Y_{k})\}_{k=1}^{\infty}\subset [0,1]$ s.t.

- For deterministic $l_{k}\to 0$ we have $0<l_{k}^{a_{1}}\leq Y_{k}-X_{k}\leq l_{k}^{a_{2}}$.
- The endpoints $\{X_{k},Y_{k}\}_{k\geq 1}$ are all correlated ,even for different k, but their lengths $|I_{k}|=Y_{k}-X_{k}, |I_{k+m}|=Y_{k+m}-X_{k+m}$ are independent when they don't intersect by a buffer: $$Y_{k+m}+\epsilon_{k+m}<X_{k}\text{ or }Y_{k}<X_{k+m}-\epsilon_{k+m},$$ where deterministic $\epsilon_{k}>0$ are decreasing to zero: $\epsilon_{k}>\epsilon_{k+1}\to 0$.

Let $a_{k,n}:=E[X_{k}X_{n}],b_{k,n}:=E[X_{k}Y_{n}],c_{k,n}:=E[Y_{k}Y_{n}]$ be the correlations.

P: Our problem is to find the probability of arranging the intervals $I_{k}$ in [0,1] so that their lengths will be independent of each other.

We are not asking for counterexamples. All we ask is whether it reminds you of some framework where this question ** might** be placed in. We will do the rest of the work of figuring it out if and how exactly. Just mention some reference and we will go through it.

**Attempts**

1)*random tree framework with factorial number of children*

The probability of the first two $|I_{1}|,|I_{2}|$ being independent (we mean their lengths) is the union of the events

$$v_{1}:=\{Y_{2}+\epsilon_{2}<X_{1}\}\text{ or }v_{2}:=\{Y_{1}<X_{2}-\epsilon_{2}\},$$

where $v_{1},v_{2}$ are the first two vertices. If $v_{1}$ occurs, then we ask for the probability that $|I_{3}|,|I_{1}|$ are independent **and** $|I_{3}|,|I_{2}|$ are independent. So now $v_{1}$ will have three children vertices $v_{1,1},v_{1,2},v_{1,3}$.

Similarly, at the kth level a vertex $v$ will have $(k+1)!$ number of children.

The problem is to find the probability of at least one infinite unary subtree i.e. at least one chain of events/vertices. That will give the probability of non-intersecting (with buffer) intervals.