Consider the cycles of the random permutation of length n. Let X n (t) be the number of cycles with length not exceeding n', t e [0,1]. The random process Y n (ί) = (X n (t) -tin n)/\rl /2 n is shown to converge weakly to the standard Brownian motion W(t),t e [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 Erdόs-Turan Theorem: it states that the group-order of random permutation is asymptotically e®, where ^is Gaussian with mean In 2 n/2 and variance In 3 n/3.

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Stochastic models of turbulent atmospheric dispersion treat either the particle displacement or particle velocity as a continuous time Markov process. An analysis of these processes using stochastic differential equation theory shows that previous particle displacement models have not correctly simulated cases in which the diffusivity is a function of vertical position. A properly formulated Markov displacement model which includes a time-dependent settling velocity, deposition and a method to simulate boundary conditions in which the flux is proportional to the concentration is presented. An estimator to calculate the mean concentration from the particle positions is also introduced. In addition, we demonstrate that for constant coefficients both the velocity and displacement models describe the same random process, but on two different time scales. The stochastic model was verified by comparison with analytical solutions of the atmospheric dispersion problem. The Monte Carlo results are in close agreement with these solutions.

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