We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence.
We then apply this coupling to resolve two conjectures of Darling and Erd\H{o}s (1956).
Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.]

Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the
classical law of the iterated logarithm of Khintchine for Brownian motion (1933).
[This portion is joint work with David Levin and Zhan Shi, and has recently appeared in
the Electr. Comm. of Probab. (2005)]