The center of mass of the ISE and the Wiener index of trees

Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $H> 1/2$

We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.

Behavior with respect to the Hurst index of the Wiener Hermite integrals and application to SPDEs

We consider the Wiener integral with respect to a $d$-parameter Hermite process with Hurst multi-index ${\bf H}= (H_{1},\ldots, H_{d}) \in \left( \frac{1}{2}, 1\right) ^{d}$ and we analyze the limit behavior in distribution of this object when the components of ${\bf H}$ tend to $1$ and/or $\frac{1}{2}$. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process.

On the asymptotic internal path length and the asymptotic Wiener index of random split trees

The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.

Distributional Itô's Formula and Regularization of Generalized Wiener Functionals

We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an Itô formula for a diffusion in a distributional setting, and apply to investigate differentiability-index $s$ and integrability-index $p \geqslant 2$ for which the Bochner integral belongs to $\mathbb{D}_{p}^{s}$.

On tail bounds for random recursive trees

We consider a multivariate distributional recursion of sum-type as arising in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

改进的复四阶矩定理

An improved complex fourth moment theorem

本文改进了复四阶矩定理的收缩条件，给出Wasserstein距离下的Berry-Esséen界，并应用于Chandler摆动模型比率过程的渐近分析。