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23
Random Walks, Capacity, and Percolation on Trees
 Ann. Probab
, 1992
"... . 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 w ..."
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Cited by 61 (9 self)
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. 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...
COHERENT MULTIPERIOD RISK ADJUSTED VALUES AND BELLMAN’S PRINCIPLE
, 2004
"... Starting with a time0 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 ..."
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Cited by 60 (6 self)
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Starting with a time0 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 decisionmaking. We finally deduce risk measurements for final value of lockedin positions and repeat a warning concerning TailValueatrisk.
The Profile of Binary Search Trees
, 2001
"... 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 ..."
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Cited by 39 (11 self)
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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...
If You’re So Smart, Why Aren’t You Rich? Belief Selection in Complete and Incomplete Markets
, 2001
"... ..."
Recursive Computation of the Invariant Distribution of a Diffusion: The Case of a Weakly Mean Reverting Drift
, 2003
"... 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 ..."
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Cited by 14 (3 self)
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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.
Martingales and large deviations for binary search trees
 Random Struct. Algorithms
, 2001
"... 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. ..."
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Cited by 12 (1 self)
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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.
Stochastic Analysis Of TreeLike Data Structures
 Proc. R. Soc. Lond. A
, 2002
"... 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. ..."
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Cited by 9 (1 self)
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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 GaltonWatson 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.
On The ErrorCorrecting Capabilities Of Cycle Codes Of Graphs
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We are interested in a function f(p) that represents the probability that a random subset of edges of a \Deltaregular 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 t ..."
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Cited by 8 (1 self)
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We are interested in a function f(p) that represents the probability that a random subset of edges of a \Deltaregular 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.
Burstiness in broadband integrated networks
 Performance Evaluation
, 1992
"... 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 th ..."
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Cited by 4 (3 self)
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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 nstage Markov Modulated Bernoulli Process, MMBPn, for short, and its peculiar case, the nstage HyperBernoulli 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.
The spread of a rumor or infection in a moving population
 2005) Shape and Propagation of Fronts 27
"... We consider the following interacting particle system: There is a “gas ” of particles, each of which performs a continuoustime simple random walk on Z d, with jump rate DA. These particles are called Aparticles and move independently of each other. They are regarded as individuals who are ignorant ..."
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Cited by 4 (2 self)
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We consider the following interacting particle system: There is a “gas ” of particles, each of which performs a continuoustime simple random walk on Z d, with jump rate DA. These particles are called Aparticles 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−) Aparticles at x, and that the NA(x,0−),x ∈ Z d, are i.i.d., meanµA Poisson random variables. In addition, there are Bparticles which perform continuoustime simple random walks with jump rate DB. We start with a finite number of Bparticles in the system at time 0. Bparticles are interpreted as individuals who have heard a certain rumor or who are infected. The Bparticles move independently of each other. The only interaction is that when a Bparticle and an Aparticle coincide, the latter instantaneously turns into a Bparticle. We investigate how fast the rumor, or infection, spreads. Specifically, if ˜B(t): = {x ∈ Z d: a Bparticle 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 Bparticles 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.