Related papers: Martingales and character ratios
A proof of the continuous martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale, after a truncation argument.
Finding upper bounds for character ratios is a fundamental problem in asymptotic group theory. Previous bounds in the symmetric group have led to remarkable applications in unexpected domains. The existing approaches predominantly relied on…
In this paper, we review the representation theory of the infinite symmetric group, and we extend the works of Kerov and Vershik by proving that the irreducible characters of the infinite symmetric group always satisfy a central limit…
Character measure is a probability measure on irreducible representations of a semisimple Lie algebra. It appears from the decomposition into irreducibles of tensor power of a fundamental representation. In this paper we calculate the…
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…
We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations, the McKay Correspondence, and Pitman's…
We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about…
We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized…
We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random variables by Maxwell and Woodroofe. Our approach is based on new results for…
We consider Jack measures on partitions with homogeneous defining specializations. For each of the six distinct classes of measures obtained this way we prove a global law of large numbers with an explicit limiting particle density. We also…
We study asymptotics of reducible representations of the symmetric groups S_q for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a…
We develop a martingale approximation approach to studying the limiting behavior of quadratic forms of Markov chains. We use the technique to examine the asymptotic behavior of lag-window estimators in time series and we apply the results…
R. Stanley has found a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, he has given a conjectural generalisation for any shape. Here, we will prove this formula…
In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character…
The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale $M$. A necessary and sufficient condition in terms of…
A stationary random sequence admits under some assumptions a representation as the sum of two others: one of them is a martingale difference sequence, and another is a so-called coboundary. Such a representation can be used for proving some…
The main result of the article is the rate of convergence to the Rosenblatt-type distributions in non-central limit theorems. Specifications of the main theorem are discussed for several scenarios. In particular, special attention is paid…
We study the convergence of a sequence of finite Blaschke products of a fix order toward a rotation. This would enable us to get a better picture of a characterization theorem for finite Blaschke products.
This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the…
This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of…