On the error bound in a combinatorial central limit theorem

Poisson--Dirichlet Limit Theorems in Combinatorial Applications via Multi-Intensities

We present new, exceptionally efficient proofs of Poisson--Dirichlet limit theorems for the scaled sizes of irreducible components of random elements in the classic combinatorial contexts of arbitrary assemblies, multisets, and selections, when the components generating functions satisfy certain standard hypotheses. The proofs exploit a new criterion for Poisson--Dirichlet limits, originally designed for rapid proofs of Billingsley's theorem on the scaled sizes of log prime factors of random integers (and some new generalizations). Unexpectedly, the technique applies in the present combinatori…

Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

Berry Esseen type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds are applied to the occurrences of fixed relatively ordered sub-sequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs on finite …

Stein's method using approximate zero bias couplings with applications to combinatorial central limit theorems under the Ewens distribution

We generalize the well-known zero bias distribution and the $λ$-Stein pair to an approximate zero bias distribution and an approximate $λ,R$-Stein pair, respectively. Berry Esseen type bounds to the normal, based on approximate zero bias couplings and approximate $λ,R$-Stein pairs, are obtained using Stein's method. The bounds are then applied to combinatorial central limit theorems where the random permutation has the Ewens $\mathcal{E}_θ$ distribution with $θ>0$ which can be specialized to the uniform distribution by letting $θ=1$. The family of the Ewens distributions appears in the context…

A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structures

A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes using independent random variables. The principle outlines conditions under which the total variation distance between the true joint distribution and the approximation should be small, and was successfully exploited by Pittel in the cases of integer partitions and set partitions. We provide sufficient conditions for this principle to be true in a general context, valid for certain discrete probability di…

Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $η\geq 0$, $\Pr(s_n(M_n) \leq η) \lesssim n^{C}η+ \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work…

On the accuracy in a combinatorial central limit theorem: the characteristic function method

The aim of this paper is to present a new proof of an explicit version of the Berry-Esséen type inequality of Bolthausen (Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66, 379--386, 1984). The literature already provides several proofs of it using variants of Stein's method. The characteristic function method has also been applied but led only to weaker results. In this paper, we show how to overcome the difficulties of this method by using a new identity for permanents of complex matrices in combination with a recently proved inequality for the characteristic function of t…

Combinatorial considerations on the invariant measure of a stochastic matrix

The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.

A combinatorial proof of Aldous-Broder theorem for general Markov chains

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph $G$, but it is more general: given an irreducible and reversible Markov chain $M$ on $G$ started at $r$, the tree rooted at $r$ formed by the first entrance steps in each node (different from the root) has a probability proportional to $\prod_{e=(e^-,e^+)\in {\sf Edges}(t,r)} M_{e^{-},e^+}$, where the edges are directed toward $r$. In this paper we give proofs of Aldous-Broder theorem in the general case, where the kernel $M$ is irreducible but not assumed to be reversible (this g…

集合划分的新组合恒等式与有限自由概率论中的极限定理

New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

本文通过建立集合划分的精细化组合恒等式，研究了有限自由概率论中的极限定理，给出了 Sakuma–Yoshida 定理的有限自由类比、有限自由乘法卷积的中心极限定理，并为 Kabluchko 的相关极限定理提供了新证明。

Esseen type bounds of the remainder in a combinatorial CLT

We derive Esseen type bounds of the remainder in a combinatorial central limit theorem for independent random variables without third moments.

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.

A combinatorial approach to a model of constrained random walkers

In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with the on…

从经典到现代的中心极限定理

From classical to modern central limit theorems

本文回顾了从De Moivre发现正态分布和经典中心极限定理，到鞅中心极限定理的历史，并介绍了非线性中心极限定理和非线性正态分布这一新研究方向。

Reversible coagulation-fragmentation processes and random combinatorial structures:asymptotics for the number of groups

We establish the central limit theorem for the number of groups at the equilibrium of a coagulation-fragmentation process given by a parameter function with polynomial rate of growth. The result obtained is compared with the one for random combinatorial structures obeying the logarithmic condition.