On The Joint Normality of Certain Statistics on Ordered Trees

广义逆序数与广义下降数的联合渐近正态性及极值

Joint Asymptotic Normality and Extremes of Generalized Inversions and Descents

本文研究经典Weyl群上广义逆序数与广义下降数的二元渐近正态性，分类渐近相关性存在性并给出公式，还推导其极值服从Gumbel分布的样本量条件。

Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton--Watson trees

We consider conditioned Galton-Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed.

由高阶矩决定的多元渐近正态性

Multivariate asymptotic normality determined by high moments

本文将高阶矩渐近行为蕴含渐近正态性的判据推广到多维情形，并以经典球箱分配中不同占用数箱子计数的联合分布为例验证其有效性。

The number of flags in finite vector spaces: Asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches infinity. Finally, we apply our statements to derive further statistical aspects of generalized Rogers-Szegoe polynomi…

On the Asymptotic Statistics of the Number of Occurrences of Multiple Permutation Patterns

We study statistical properties of the random variables $X_σ(π)$, the number of occurrences of the pattern $σ$ in the permutation $π$. We present two contrasting approaches to this problem: traditional probability theory and the ``less traditional'' computational approach. Through the perspective of the first one, we prove that for any pair of patterns $σ$ and $τ$, the random variables $X_σ$ and $X_τ$ are jointly asymptotically normal (when the permutation is chosen from $S_{n}$). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normalit…

Exploring hypergraphs with martingales

Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-Löf, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper "Counting connected hypergraphs via the probabilistic met…