. A collection of several different probabilistic processes involving trees is shown to have an unexpected unity. This makes possible a fruitful interplay of these probabilistic processes. The processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well. Our work has five specific aims: First, an exact correspondence between random walks and percolation on trees is proved, extending and sharpening previous work of the author. This is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes. Second, we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation. This capacitary formulation extends and sharpens work of Fan on random interval coverings. Third, we show how this formulation also applies to generalize work of Evans on random labelling of trees. Fourth, the correspondence between random walks and percolation is used...

Starting with a time-0 coherent risk measure defined for “value processes”, we define, also at intermediate times, a risk measurement process. Two other constructions of such measurement processes are given in terms of sets of test probabilities. These constructions are identical and related to the former construction when the sets fulfill a stability condition also met in multiperiod treatment of ambiguity in decision-making. We finally deduce risk measurements for final value of locked-in positions and repeat a warning concerning Tail-Value-at-risk.

We characterize the limiting behaviour of the number of nodes in level k of binary search trees Tn in the central region 1:2 log n k 2:8 log n. Especially we show that the width V n (the maximal number of internal nodes at the same level) satises V n (n= p 4 log n) as n !1 a.s. 1 Introduction A binary search tree is a binary tree, in which each (internal) node is associated to a key, where the keys are drawn from some totally ordered set, say f1; 2; : : : ; ng. The rst key is associated to the root. Now, the next key is put to the left child of the root if it is smaller than the key of the root and it is put to the right child of the root if it is larger than the key of the root. In this way we proceed further by inserting key by key. So starting from a permutation of f1; 2; : : : ; ng we get a binary tree with n (internal) nodes such that the keys of the left subtree of any given node x are smaller than the key of x and the keys of the right subtree are larger than the...

Abstract not found

We establish an almost sure large deviations theorem for the depth of the external nodes of binary search trees (BST). To achieve this, a parametric family of martingales is introduced. This family also allows us to get asymptotic results on the number of external nodes at deepest level.

The purpose of this article is to present two types of data structures, binary search trees and usual (combinatorial) binary trees. Although they constitute the same set of (rooted) trees they are constructed via completely dierent rules and thus the underlying probabilitic models are dierent, too. Both kinds of data structures can be analyzed by probabilistic and stochastic tools, binary search trees (more or less) with martingales and binary trees (which can be considered as a special case of Galton-Watson trees) with stochastic processes. It is also an aim of this article to demonstrate the strength of analytic methods in speci c parts of probabilty theory related to combinatorial problems, especially we make use of the concept of generating functions. One reason is that that recursive combinatorial descriptions can be translated to relations for generating functions, and second analytic properties of these generating functions can be used to derive asymptotic (probabilistic) relations. 1.

We are interested in a function f(p) that represents the probability that a random subset of edges of a \Delta-regular graph G contains half the edges of some cycle of G. f(p) is also the probability that a codeword be corrupted beyond recognition when words of the cycle code of G are submitted to the binary symmetric channel. We derive a precise upperbound on the largest p for which f(p) can vanish when the number of edges of G goes to infinity. To this end we introduce the notion of fractional percolation on trees, and calculate the related critical probabilities.

We study a recursive procedure, based on the Euler scheme with decreasing step, for the computation of the invariant distribution of a Brownian di#usion process satisfying weak stability conditions. We are able to extend some of the results of [4] to di#usions for which the drift size at infinity is very small.

Since broadband integrated networks have to cope with a wide range of bit rates, the notion of burstiness which expresses the irregularity ofa ow, has been recognized as a vital question for such networks. In burstiness characterization encountered in the literature, special attention is given to the squared coe cient ofvariation of interarrival time (Cv 2) in a cell arrival process. In order to observe the impact of a bursty ow on a queue, we introduce in this paper a new class of arrival process, the n-stage Markov Modulated Bernoulli Process, MMBPn, for short, and its peculiar case, the n-stage Hyper-Bernoulli process, denoted by HBPn. Wenumerically solve the MMBPn=D=1=K and we compute in particular the rejection probability and the mean waiting time. For that purpose, a relation between the stationary queue length distribution and arrival time distribution is established. This relation adapts the GASTA equality to the arrival process under consideration. We then discuss the relevance of Cv 2 for burstiness characterization through an example: the HBP 2=D=1=K queue. We show that Cv 2 becomes signi cant only when local overload occurs, i.e., when the arrival rate is momentarily greater than the server rate. The results are then applied to two basic ATM problems: tra c characterization and bu er dimensioning using bursty inputs.

We consider the following interacting particle system: There is a “gas ” of particles, each of which performs a continuous-time simple random walk on Z d, with jump rate DA. These particles are called A-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with NA(x,0−) A-particles at x, and that the NA(x,0−),x ∈ Z d, are i.i.d., mean-µA Poisson random variables. In addition, there are B-particles which perform continuous-time simple random walks with jump rate DB. We start with a finite number of B-particles in the system at time 0. B-particles are interpreted as individuals who have heard a certain rumor or who are infected. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if ˜B(t): = {x ∈ Z d: a B-particle visits x during [0,t]} and B(t) = ˜B(t)+[−1/2,1/2] d, then we investigate the asymptotic behavior of B(t). Our principal result states that if DA = DB (so that the A- and B-particles perform the same random walk), then there exist constants 0 < Ci < ∞ such that almost surely C(C2t) ⊂ B(t) ⊂ C(C1t) for all large t, where C(r) = [−r,r] d. In a further paper we shall use the results presented here to prove a full “shape theorem, ” saying that t −1 B(t) converges almost surely to a nonrandom set B0, with the origin as an interior point, so that the true growth rate for B(t) is linear in t. If DA ̸ = DB, then we can only prove the upper bound B(t) ⊂ C(C1t) eventually.