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Search Results: 1 - 10 of 14797 matches for " Christian Houdré "
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Concentration of the Spectral Measure for Large Random Matrices with Stable Entries
Christian Houdré,Hua Xu
Mathematics , 2007,
Abstract: We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
On the Limiting Shape of Young Tableaux Associated With Inhomogeneous Random Words
Christian Houdré,Hua Xu
Mathematics , 2009,
Abstract: The limiting shape of the random Young diagrams associated with an inhomogeneous random word is identi?ed as a multidimensional Brownian functional. This functional is identical in law to the spectrum of a random matrix. The Poissonized word problem is also briefy studied, and the asymptotic behavior of the shape analyzed.
Simultaneous large deviations for the shape of Young diagrams associated with random words
Christian Houdré,Jinyong Ma
Mathematics , 2011, DOI: 10.3150/14-BEJ612
Abstract: We investigate the large deviations of the shape of the random RSK Young diagrams associated with a random word of size $n$ whose letters are independently drawn from an alphabet of size $m=m(n)$. When the letters are drawn uniformly and when both $n$ and $m$ converge together to infinity, $m$ not growing too fast with respect to $n$, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. In the non-uniform case, a control of both highest probabilities will ensure that the length of the top row of the diagram satisfies a large deviation principle. In either case, both speeds and rate functions are identified. To complete our study, non-asymptotic concentration bounds for the length of the top row of the diagrams, that is, for the length of the longest increasing subsequence of the random word are also given for both models.
On the Order of the Central Moments of the Length of the Longest Common Subsequence
Christian Houdré,Jinyong Ma
Mathematics , 2012,
Abstract: We investigate the order of the $r$-th, $1\le r < +\infty$, central moment of the length of the longest common subsequence of two independent random words of size $n$ whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, a lower bound is shown to be of order $n^{r/2}$. This result complements a generic upper bound also of order $n^{r/2}$.
On the Variance of the Optimal Alignment Score for an Asymmetric Scoring Function
Christian Houdré,Heinrich Matzinger
Mathematics , 2007,
Abstract: We investigate the variance of the optimal alignment score of two independent iid binary, with parameter 1/2, sequences of length $n$. The scoring function is such that one letter has a somewhat larger score than the other letter. In this setting, we prove that the variance is of order $n$, and this confirms Waterman's conjecture in this case.
Viscosity and Principal-Agent Problem
Ruoting Gong,Christian Houdré
Mathematics , 2009,
Abstract: We develop a stochastic control system from a continuous-time Principal-Agent model in which both the principal and the agent have imperfect information and different beliefs about the project. We consider the agent's problem in this stochastic control system, i.e., we attempt to optimize the agent's utility function under the agent's belief. Via the corresponding Hamilton-Jacobi-Bellman equation the value function is shown to be jointly continuous and to satisfy the Dynamic Programming Principle. These properties directly lead to the conclusion that the value function is a viscosity solution of the HJB equation. Uniqueness is then also established.
On the rate of approximation in finite-alphabet longest increasing subsequence problems
Christian Houdré,Zsolt Talata
Mathematics , 2009, DOI: 10.1214/12-AAP853
Abstract: The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of $\log n/\sqrt{n}$ is obtained. The uniform binary case is further explored, and an improved $1/\sqrt{n}$ rate obtained.
A probabilistic approach to the asymptotics of the length of the longest alternating subsequence
Christian Houdré,Ricardo Restrepo
Mathematics , 2010,
Abstract: Let $LA_{n}(\tau)$ be the length of the longest alternating subsequence of a uniform random permutation $\tau\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(\tau)$. Our methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences in which case our results are mainly original. A sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is finally presented.
On the Limiting Shape of Young Diagrams Associated With Markov Random Words
Christian Houdré,Trevis J. Litherland
Mathematics , 2011,
Abstract: Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space an ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by showing that, under a cyclic condition, a spectral characterization of the Markov transition matrix delineates precisely when the limiting shape is the spectrum of the $m \times m$ traceless GUE. For $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously only known for $m=2$. However, this is no longer true for $m \ge 4$. In arbitrary dimension, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. To finish, we explore, in this general setting, connections between various limiting laws and spectra of Gaussian random matrices, focusing in particular on the relationship between the terminal points of the Brownian motions, the diagonals of the random matrix, and the scaling of the off-diagonals, a scaling we conjecture to be a function of the spectrum of the covariance matrix governing the Brownian motion.
Asymptotics for the Length of the Longest Increasing Subsequence of Binary Markov Random Word
Christian Houdré,Trevis J. Litherland
Mathematics , 2011,
Abstract: Let $(X_n)_{n\ge 0}$ be an irreducible, aperiodic, and homogeneous binary Markov chain and let $LI_n$ be the length of the longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$. Using combinatorial constructions and weak invariance principles, we present elementary arguments leading to a new proof that (after proper centering and scaling) the limiting law of $LI_n$ is the maximal eigenvalue of a $2 \times 2$ Gaussian random matrix. In fact, the limiting shape of the RSK Young diagrams associated with the binary Markov random word is the spectrum of this random matrix.
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