What does this mean $[X]_1^T$?
This notation is quite common in EE works: 
If $S=S_1S_2S_3\cdots$ is a string (of some length), then

$S_i^j$ is the substring between places $i$ and $j$, that is, $S_i^j=S_i\cdots S_j$.

I didn't looks at the paper, but I guess that $[X]$ is a string (of expectations? of $X$ directly? this should be defined in the paper), and they just look at a prefix of length $T$ of that string.