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.

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.

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.

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 $n^{r/2}$.

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.

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.

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.

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.

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.

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.