  It is known that after scaling a random Motzkin path converges to a Brownian
excursion. We prove that the fluctuations of the counting processes of the
ascent steps, the descent steps and the level steps converge jointly to linear
combinations of two independent processes: a Brownian motion and a Brownian
excursion. The proofs rely on the Laplace transforms and an integral
representation based on an identity connecting non-crossing pair partitions and
joint moments of an explicit non-homogeneous Markov process.
