Central Limit Theorems are known for the Eulerian statistic "descent" (or "excedance") in the symmetric group \(\SSS_n\). Recently, Fulman, Kim, Lee and Petersen gave a Central Limit Theorem for "descent" over the alternating group \(\AAA_n\) and also gave a Carlitz identity in \(\AAA_n\) using descents. In this paper, we give a Central Limit Theorem in \(\AAA_n\) involving excedances. We extend these to the positive elements in type B and type D Coxeter groups. Boroweic and M\l{}otkowski enumerated type B descents over \(\DD_n\), the type D Coxeter group and gave similar results. We refine their results for both the positive and negative part of \(\DD_n\). Our results are a consequence of signed enumeration over these subsets.
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