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Central Limit Theorems are known for the Eulerian statistic "descent" (or"excedance") in the symmetric group $\SSS_n$. Recently, Fulman, Kim, Lee andPetersen 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 $\AAA_n$ involvingexcedances. We extend these to the positive elements in type B and type DCoxeter groups. Boroweic and M\l{}otkowski enumerated type B descents over$\DD_n$, the type D Coxeter group and gave similar results. We refine theirresults for both the positive and negative part of $\DD_n$. Our results are aconsequence of signed enumeration over these subsets.

Eulerian Central Limit Theorems and Carlitz identities in positive elements of Classical Weyl Groups