The distribution of $T$ is indeed asymptotically normal, by virtue of any one of the many central theorems for stationary weakly dependent (here, even $2$-independent) random variables. See e.g. Theorem 0 in [Bradley][1] (due to Ibragimov), the conditions of which are easily verified in your case. [1]: http://www.sciencedirect.com/science/article/pii/0047259X81901287/pdf?md5=79d9e376f3e4842f970ebbbdcbeecffe&pid=1-s2.0-0047259X81901287-main.pdf