Consider the cycles of the
random permutation of length n. Let Xn(t) be the number of cycles with length not
exceeding nt, t ∈ [0,1]. The random process Yn(t) = (Xn(t) − tlnn)∕ln1∕2n is
shown to converge weakly to the standard Brownian motion W(t), t ∈ [0,1]. It
follows that, as a process, the empirical distribution function of “loglengths” of the
cycles weakly converges to the Brownian Bridge process. As another application, an
alternative proof is given for the Erdös-Turán Theorem: it states that the
group-order of random permutation is asymptotically e𝒰, where 𝒰 is Gaussian with
mean ln2n∕2 and variance ln3n∕3.
