In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

We consider the coupon collection problem, where each coupon is one of the types 1,…,s with probabilities given by a vector 𝒑. For specified numbers r 1,…,r s , we are interested in finding 𝒑 that minimizes the expected time to obtain at least r i type-i coupons for all i=1,…,s. For example, for s=2, r 1=1, and r 2=r, we show that p 1=(logr−log(logr))∕r is close to optimal.

We consider an r component system having an arbitrary binary monotone structure function. We suppose that shocks occur according to a point process and that, independent of what has already occurred, each new shock is one of r different types, with respective probabilities p1, …, pr. We further suppose that there are given integers n1, …, nr such that component i fails (and remains failed) when there have been a total of ni type-i shocks. Letting L be the time at which the system fails, we are interested in using simulation to estimate E[L], E[L2], and P(L > t). We show how to efficiently accomplish this when the point process is (i) a Poisson, (ii) a renewal, and (iii) a Hawkes process.

Radix Sort is a sorting algorithm based on analyzing digital data. We study the number of swaps made by Radix Select (a one-sided version of Radix Sort) to find an element with a randomly selected rank. This kind of grand average provides a smoothing over all individual distributions for specific fixed-order statistics. We give an exact analysis for the grand mean and an asymptotic analysis for the grand variance, obtained by poissonization, the Mellin transform, and depoissonization. The digital data model considered is the Bernoulli(p). The distributions involved in the swaps experience a phase change between the biased cases (p ≠ ½) and the unbiased case (p = ½). In the biased cases, the grand distribution for the number of swaps (when suitably scaled) converges to that of a perpetuity built from a two-point distribution. The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. In the unbiased case the same scaling for the number of swaps gives a limiting constant in probability.

We investigate the average similarity of random strings as captured by the average number of ‘cousins’ in the underlying tree structures. Analytical techniques including poissonization and the Mellin transform are used for accurate calculation of the mean. The string alphabets we consider are m-ary, and the corresponding trees are m-ary trees. Certain analytic issues arise in the m-ary case that do not have an analog in the binary case.

The trie is a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

Consider the classical coupon-collector's problem in which items of m distinct types arrive in sequence. An arriving item is installed in system i ≥ 1 if i is the smallest index such that system i does not contain an item of the arrival's type. We study the expected number of items in system j at the moment when system 1 first contains an item of each type

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.