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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 74 (8 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.
If You’re So Smart, Why Aren’t You Rich? Belief Selection in Complete and Incomplete Markets
, 2001
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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 41 (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...
Recursive computation of the invariant distribution of a diffusion: The case of a weakly mean reverting drift
 Stoch. Dynamics
, 2003
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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 15 (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 10 (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 on graphs
 Combinatorics, Probability and Computing
, 1997
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THEOREME DE DONSKER ET FORMES DE DIRICHLET
, 2006
"... Abstract. We use the language of errors to handle local Dirichlet forms with squared field operator (cf [2]). Let us consider, under the hypotheses of Donsker theorem, a random walk converging weakly to a Brownian motion. If, in addition, the random walk is supposed to be erroneous, the convergence ..."
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Cited by 5 (5 self)
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Abstract. We use the language of errors to handle local Dirichlet forms with squared field operator (cf [2]). Let us consider, under the hypotheses of Donsker theorem, a random walk converging weakly to a Brownian motion. If, in addition, the random walk is supposed to be erroneous, the convergence occurs in the sense of Dirichlet forms and induces the OrnsteinUhlenbeck structure on the Wiener space. This quite natural result uses an extension of Donsker theorem to functions with quadratic growth. As an application we prove an invariance principle for the gradient of the maximum of the Brownian path computed by Nualart and Vives. Résumé. Nous employons le langage des erreurs pour manier les formes de Dirichlet locales avec carré du champ (cf [2]). Considérant une promenade aléatoire convergeant en loi vers un mouvement brownien sous les hypothèses du théorème de Donsker, nous montrons que si la promenade est supposée de plus erronée, la convergence a lieu au sens des formes de Dirichlet et induit la structure d’OnsteinUhlenbeck sur l’espace de Wiener. Ce résultat bien naturel nécessite l’extension du théorème de Donsker aux fonctions à croissance quadratique. A titre d’application nous en déduisons un principe d’invariance pour le gradient du maximum de la courbe brownienne calculé par Nualart et Vives. Mots clés: promenade aléatoire, mouvement brownien, gradient, forme de Dirichlet, erreur.
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 5 (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.