We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the m-th smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the m-th smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 -m)H n+1-m ) , n, where Hm = 1#k#m k -1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...

Abstract The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ∩[0, 1] for a self-similar random set S ⊂ R+ are those which are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of n is defined by the first n terms of a random binary sequence of infinite length. The locations of 1s in the sequence are the places visited by an increasing time-homogeneous Markov chain on the positive integers if and only if S = exp(−W) for some stationary regenerative random subset W of the real line. Complementing our study in previous papers, we identify self-similar Markovian composition structures associated with the two-parameter family of partition structures. 1

Abstract. We consider linear iterated function systems with a random mul-tiplicative error on the real line. Our system is {x ↦ → di + λiY x} m i=1, where di ∈ R and λi> 0 are fixed and Y> 0 is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of i.i.d. errors y1, y2,..., dis-tributed as Y, independent of everything else. Let h be the entropy of the process, and let χ = E [log(λY)] be the Lyapunov exponent. Assuming that χ < 0, we obtain a family of conditional measures νy on the line, parametrized by y = (y1, y2,...), the sequence of errors. Our main result is that if h> |χ|, then νy is absolutely continuous with respect to the Lebesgue measure for a.e. y. We also prove that if h < |χ|, then the measure νy is singular and has di-mension h/|χ | for a.e. y. These results are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class of random sets considered by R. Arratia, motivated by probabilistic number theory. 1.

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/nζ where G is a “generator ” matrix, that is G(i,j)> 0 for i,j distinct, and G(i,i) = − ∑ k̸=i G(i,k), and ζ> 0 is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit some different, perhaps unexpected, occupation behaviors depending on parameters. Although it is shown, on the one hand, that the position at time n converges to a point-mixture for all ζ> 0, on the other hand, the average occupation vector up to time n, when variously 0 < ζ < 1, ζ> 1 or ζ = 1, is seen to converge to a constant, a pointmixture, or a distribution µG with no atoms and full support on a simplex respectively, as n ↑ ∞. This last type of limit can be interpreted as a sort of “spreading ” between the cases 0 < ζ < 1 and ζ> 1. In particular, when G is appropriately chosen, intriguingly, µG is a Dirichlet distribution, reminiscent of results in Pólya urns. Research supported in part by nsa-h982300510041 and NSF-DMS-0504193 Key words and phrases: laws of large numbers, nonhomogeneous, Markov, occupation, reinforcement, Dirichlet distribution.