We have two copied sequences of correlated continuous positive random variables that are independent of each other $(X_{n})\perp(Y_{n})$ and equal in distribution $X_{n}\stackrel{dis}{=}Y_{n}$ for each $n\geq 1$ (and with the same correlation structure too so $(X_{n})\stackrel{dis}{=}(Y_{n})$).
(Also, both converge to zero as $n\to +\infty$). We have some more information about their distribution and correlation structure. We are trying to see if we can extract a large common subsequence that are close $$|X_{n_{k}}-Y_{n_{k}}|<\delta_{n_{k}},$$
for ,ideally, very small $\delta_{n_{k}}\to 0$ that is as sharp as possible (it should be smaller than just bounding by the variables themselves that go to zero, so maybe a natural assumption is $\delta_{n}<2E[X_{n}]$). In particular, it would be great to get the following deviation
$$\mathbb{P}\left[\sum_{k=1}^{N} 1\{ |X_{k}-Y_{k}|<\delta_{k}\}<\alpha N \right ]\leq c_{1}e^{-c_{2}N},$$ for constants $\alpha,c_{1},c_{2}>0$.
Any recommendations on articles or approaches will be much appreciated (so that we can see if our particular variables satisfy the assumptions from the literature because as Quas suggested more assumptions will probably be needed).
For the case when $X_{n},Y_{n}$ are integer-valued, there are lots of results (under the keyword "longest common subsequence"). So my guess that my best bet would be to take some discrete approximation and go from there.
Another possibility, would be to study random interval graphs, (vertices are random intervals with iid endpoints and the edges are present when two such intervals intersect). So then here I suppose one can study the random intervals $I_{i}=[-\delta_{i}+X_{i}, X_{i}+\delta_{i}]$ and $J_{i}=[-\delta_{i}+Y_{i}, Y_{i}+\delta_{i}]$ and see if they intersect. But the technology developed so far is mostly about iid endpoints.
