Start with a necklace consisting of one white bead and one black bead, and add new beads one at a time by inserting each new bead between a randomly chosen adjacent pair of old beads, with the proviso that the new bead will be white if and only if both beads of the adjacent pair are black. Let Wn denote the number of white beads when the total number of beads is n. We show that EWn = n/3 and, with c2 = 2/45, that (Wn − n/3) / c√n is asymptotically standard normal. We find that, for all r ≥ 1 and n > 2r, the rth cumulant of the distribution of Wn is of the form nhr. We find the expected numbers of gaps of given length between white beads, and examine the asymptotics of the longest gaps.

A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.

We consider an independent long-range bond percolation on Z2. Horizontal and vertical bonds of length n are independently open with probability p_n ∈ [0, 1]. Given ∑n=1∞∏i=1n(1 − pi) < ∞, we prove that there exists an infinite cluster of open bonds of length less than or equal to N for some large but finite N. The result gives a partial answer to the truncation problem.