Random Forest proof notation

Question

I am having a bit of difficulty understanding the notation in equation (1) on page 4 of the following paper:

https://escholarship.org/uc/item/35x3v9t4#page-4

Specifically, what do $E_{X,Y}$ and $E_\theta$ mean? It seems like it is a saying the $PE$ of the forest tends toward the $PE$ of a single tree as $k \to \infty$, but that doesn't make sense.

Answer 1

$E_{X,Y} $ is the expectation over all X and Y. Similarly for $E_\theta$. What equation(1) says is as $k$ increase, the average of all trees is getting closer to the expectation of the trees.