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In this note, by employing a nice property of semicircular distributions, we derive some identities for the Narayana polynomial and its derivatives., Comment: 5 pages

We develop a general method to study the Fisher information distance in central limit theorem for nonlinear statistics. We first construct completely new representations for the score function. We then use these representations to derive quantitative estimates for the Fisher information distance. To illustrate the applicability of our approach, explicit rates of Fisher information convergence for quadratic forms and the functions of sample means are provided. For the sums of independent random variables, we obtain the Fisher information bounds without requiring the finiteness of Poincar\'e constant. Our method can also be used to bound the Fisher information distance in non-central limit theorems., Comment: Final version. To appear in PTRF

Most of the current task-oriented dialogue systems (ToD), despite having interesting results, are designed for a handful of languages like Chinese and English. Therefore, their performance in low-resource languages is still a significant problem due to the absence of a standard dataset and evaluation policy. To address this problem, we proposed ViWOZ, a fully-annotated Vietnamese task-oriented dialogue dataset. ViWOZ is the first multi-turn, multi-domain tasked oriented dataset in Vietnamese, a low-resource language. The dataset consists of a total of 5,000 dialogues, including 60,946 fully annotated utterances. Furthermore, we provide a comprehensive benchmark of both modular and end-to-end models in low-resource language scenarios. With those characteristics, the ViWOZ dataset enables future studies on creating a multilingual task-oriented dialogue system.

In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete K\"{a}hler manifolds. We also give several applications to study non-compact K\"{a}hler manifolds with parallel Bochner tensor or quaternion K\"{a}hler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact K\"{a}hler manifolds., Comment: arXiv admin note: text overlap with arXiv:2111.07496; text overlap with arXiv:2105.09290 by other authors, to appear in Journal of Mathematical Analysis and Applications

In this paper, we show several rigidity results of harmonic tensors in terms of Lichnerowicz Laplacians. These are obtained mainly from the method of controlling the curvature term of Lichnerowicz Laplacians due to P. Petersen and M. Wink in \cite{PW21}. Geometric applications on several classes of Riemannian manifolds including Einstein manifolds and immersed submanifolds are provided., Comment: We have removed Theorem 1.3 and modified Theorem 1.4 in the previous version and combine them as Theorem 1.3 in this version. Any comments are welcome

In this note, we investigate the density of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide a log-normal upper bound for the density., Comment: 9 pages,to appear in Comptes Rendus Mathematique

In this note, we revisit a classical problem related to the density of nonlinear statistics. We obtain a new representation of densities and, for the first time, a necessary and sufficient condition for the existence of densities is provided., Comment: 8 pages

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.

We prove a Yau's type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry-Emery Ricci tensor and the weighted mean curvature are bounded below. As an application, we give a Liouville type result for bounded $f$-harmonic functions with the Dirichlet boundary condition. Our results do not depend on any assumption on the potential function $f$., Comment: 10 pages, 1 figure,final version, accepted by Nonlinear Analysis

We develop a novel stochastic primal dual splitting method with Bregman distances for solving a structured composite problems involving infimal convolutions in non-Euclidean spaces. The sublinear convergence in expectation of the primal-dual gap is proved under mild conditions on stepsize for the general case. The linear convergence rate is obtained under additional condition like the strong convexity relative to Bregman functions.

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate $\mathcal{O}(log(n)/n)$ in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex-concave saddle point problems, we derive the rate of the primal-dual gap. In particular, we also obtain $\mathcal{O}(1/n)$ rate convergence of the primal-dual gap in the deterministic setting.