Let [An, Bn] be random subintervals of [0, 1] defined recursively as follows. Let A1 = 0, B1 = 1 and take Cn, Dn to be the minimum and maximum of k, i.i.d. random points uniformly distributed on [An, Bn]. Choose [An+1, Bn+1] to be [Cn, Bn], [Any Dn] or [Cn, Dn] with probabilities p, q, r respectively, p + q + r = 1. It is shown that the limiting distribution of [Any Bn] has the beta distribution on [0,1] with parameters k(p + r) and k(q + r). The result is used to consider a randomized version of Golden Section search.
