9502325T Faculty of Science School of Mathematics / Abstract Increasing trees are labelled rooted trees in which labels along any branch from the root appear in increasing order. They have numerous applications in tree representations of permutations, data structures in computer science and probabilistic models in a multitude of problems. We use a generating function approach for the computation of parameters arising from such trees. The generating functions for some parameters are shown to be related to ordinary differential equations. Singularity analysis is then used to analyze several parameters of the trees asymptotically.Various classes of trees are considered. Parameters such as depth and path length for heap ordered trees have been analyzed in [35]. We follow a similar approach to determine grand averages for such trees. The model is that p of the n nodes are labelled at random in 􀀀n p(ways, and the characteristic parameters depend on these labelled nodes. Also, we will attempt to look at the limiting distributions involved. Often, when they are Gaussian, Hwang's quasi power theorem, from [18], can be employed. Spanning tree size and the Wiener index for binary search trees have been computed in [33]. The Wiener index is the sum of all distances between pairs of nodes in a tree. Arelated parameter of interest is the Steiner distance which generalises, to sets of k nodes, the Wiener index (k=2). Furthermore, the distribution of the size of the ancestor-tree and of the induced spanning subtree for random trees is presented in [30]. The same procedure is followed to obtain the Steiner distance for heap ordered trees and for other varieties of increasing trees.

We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a (sub-)linear function of the indegree of the older vertex at that time. We provide a limit theorem with rates of convergence for the distribution of a vertex, chosen uniformly at random, as the number of vertices tends to infinity. To do so, we develop Stein's method for a new class of limting distributions including power-laws. Similar, but slightly weaker results are shown to be deducible using coupling techniques. Concentrating on a specific preferential attachment model we also show that the outdegree distribution asymptotically follows a Poisson law. In addition, we deduce a central limit theorem for the number of isolated vertices. We thereto construct a size-bias coupling which in combination with Stein’s method also yields bounds on the distributional distance.

<div>In this thesis we study two different types of self-interacting random walks. First, we study excited random walk in a deterministic, identically-piled cookie environment under the constraint that the total drift contained in the cookies at each site is finite. We show that the walk is recurrent when this parameter is between -1 and 1 and transient when it is less than -1 or greater than 1. In the critical case, we show that the walk is recurrent under a mild assumption on the environment. We also construct an environment where the total drift per site is 1 but in which the walk is transient. This behavior was not present in previously-studied excited random walk models.</div><div><br></div><div>Second, we study the "have your cookie and eat it'' random walk proposed by Pinsky, who already proved criteria for determining when the walk is recurrent or transient and when it is ballistic. We establish limiting distributions for both the hitting times and position of the walk in the transient regime which, depending on the environment, can be either stable or Gaussian.</div>

Jiang (1997) 首先提出多變量d轉換與其性質。利用多變量d轉換，我們可以定義新式的特徵函數，並且稱它們是多變量d特徵函數。在這篇論文中，我們將使用多變量d特徵函數來證明在普通的條件下，Dirichlet隨機向量的線性組合會分配收斂（converge in distribution）到一個對稱的分配。此外，當給定一個分配函數的多變量d特徵函數，我們將建構一個方法來決定此分配函數。另一方面，我們將證明多變量d特徵函數擁有很多類似傳統的特徵函數的性質。 / A multivariate d-transformation and its properties were first given by Jiang (1997). By means of the multivariate d-transformations, we can define new kinds of characteristic functions and call them multivariate d-characteristic functions. In this thesis, we will use the multivariate d-characteristic function to show that the linear combinations of Dirichlet random vectors, under regularity conditions, converge in distribution to a spherical distribution. Moreover, We will construct a method for constructing the distribution function with a given multivariate d-characteristic function. In addition, we will show that the multivariate d-characteristic function has many properties which are similar to those of the traditional characteristic function.