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246
The objective method: Probabilistic combinatorial optimization and local weak convergence
, 2003
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Functional Limit Theorems For Multitype Branching Processes And Generalized Pólya Urns
 APPL
, 2004
"... A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the ..."
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Cited by 106 (16 self)
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A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 102 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
On Convergence Rates in the Central Limit Theorems for Combinatorial Structures
, 1998
"... Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This th ..."
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Cited by 82 (11 self)
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Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.
Varieties of Increasing Trees
, 1992
"... An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We ..."
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Cited by 80 (7 self)
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An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.
A general limit theorem for recursive algorithms and combinatorial structures
 ANN. APPL. PROB
, 2004
"... Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer ..."
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Cited by 76 (26 self)
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Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and to use the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common ℓ2 metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or mary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.
Asymptotic Behavior of the LempelZiv Parsing Scheme and Digital Search Trees
 Theoretical Computer Science
, 1995
"... The LempelZiv parsing scheme finds a wide range of applications, most notably in data compression and algorithms on words. It partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phrase. The parameter of interest is the n ..."
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Cited by 72 (34 self)
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The LempelZiv parsing scheme finds a wide range of applications, most notably in data compression and algorithms on words. It partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phrase. The parameter of interest is the number M n of phrases that one can construct from a sequence of length n. In this paper, for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of M n which turns out to be normal. This proves a long standing open problem. In fact, to obtain this result we solved another open problem, namely, that of establishing the limiting distribution of the internal path length in a digital search tree. The latter is a consequence of an asymptotic solution of a multiplicative differentialfunctional equation often arising in the analysis of algorithms on words. Interestingly enough, our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability, and analytical techniques such as Mellin transform, differentialfunctional equations, dePoissonization, and so forth. In concluding remarks we indicate a possibility of extending our results to Markovian models.
Probability Distributions on Cladograms
 In Random Discrete Structures
, 1996
"... By analogy with the theory surrounding the Ewens sampling formula in neutral population genetics, we ask whether there exists a natural oneparameter family of probability distributions on cladograms ("evolutionary trees") which plays a central role in neutral evolutionary theory. Unfortuna ..."
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Cited by 66 (2 self)
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By analogy with the theory surrounding the Ewens sampling formula in neutral population genetics, we ask whether there exists a natural oneparameter family of probability distributions on cladograms ("evolutionary trees") which plays a central role in neutral evolutionary theory. Unfortunately the answer seems to be "no"  see Conjecture 2. But we can embed the two most popular models into an interesting family which we call "betasplitting" models. We briefly describe some mathematical results about this family, which exhibits qualitatively different behavior for different ranges of the parameter fi. 1 Probability distributions on partitions and neutral population genetics The first few sections give some conceptual background. The reader wishing to "get right to the point" should skim these and proceed to section 3. For each n there is a finite set of partitions of f1; 2; : : : ; ng into unordered families fA 1 ; A 2 ; : : : ; A k g of subsets. A oneparameter family (P (n) ` ) o...