  In this paper we use a probabilistic approach to derive the expressions for
the characteristic functions of basic statistics defined on permutation
tableaux. Since our expressions are exact, we can identify the distributions of
basic statistics (like the number of unrestricted rows, the number of rows, and
the number of 1s in the first row) exactly. In all three cases the
distributions are known to be asymptotically normal after a suitable
normalization. We also establish the asymptotic normality of the number of
superfluous 1s. The latter result relies on a bijection between permutation
tableaux and permutations and on a rather general sufficient condition for the
central limit theorem for the sums of random variables in terms of dependency
graph of the summands.
