Maximum and minimum of one-dimensional diffusions

Let M_t be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that |P(M_t?x) ? F^t(x)|→0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the maximum of independent and identically distributed random variables with distribution function F. A new proof of this fact is given which is based on a time change of the Ornstein-Uhlenbeck process. Using this technique, the asymptotic independence of the maximum and minimum is also established. Moreover, this method allows one to construct stationary processes in which the limiting behavior of M_t is essentially unaffected by the stationary distribution. That is, there may be no relationship between the distribution F above and the marginal distribution of the process.