Probabilities of randomly centered small balls and quantization in Banach spaces

The Asymptotic Distribution of Randomly Weighted Sums and Self-normalized Sums

We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where ${Y_{i} : i\geq 1}$ are non-negative i.i.d. random variables, and ${X_{i} : i\geq 1} $ are i.i.d. random variables, independent of ${Y_{i} : i \geq 1}$. The main result of the paper is that each subsequential limit law of T_n$ is continuous for any non-degenerate $X_1$ with finite expectation, if and only if $Y_1$ is in the centered Feller class.

对数相关高斯场中心化最大值的收敛性

Convergence of the centered maximum of log-correlated Gaussian fields

本文证明任意维数下对数相关高斯场的中心化最大值在协方差适当收敛时依分布收敛到随机平移的Gumbel分布，并刻画了随机平移的极限结构。

Appendix To Limits For Partial Maxima Of Gaussian Random Vectors

This appendix provides a short proof for sample path continuity of the Brownian motion induced by an arbitrary centered Gaussian measure on a separable Banach space, and also some perturbation results for the spectrum of compact self-adjoint operators on a Hilbert space.

Banach空间值随机特征模型的通用逼近性质，包括随机神经网络

Universal approximation property of Banach space-valued random feature models including random neural networks

本文研究Banach空间值随机特征学习的通用逼近性质，通过随机初始化特征映射仅训练线性读出层，证明在Bochner空间中的逼近结果，并推导逼近速率和学习算法。

Banach空间中局部平稳时间序列的函数中心极限定理

A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces

本文为Banach空间中的局部平稳时间序列建立了函数中心极限定理，不依赖type-2或cotype假设，并应用于自归一化CUSUM变点检测。

Infinite dimensional reflecting Ornstein-Uhlenbeck stochastic process

In this article we introduce the Gaussian Sobolev space $W^{1,2}(\mathscr O,γ)$, where $\mathscr O$ is an arbitrary open set of a separable Banach space $E$ endowed with a nondegenerate centered Gaussian measure $γ$. Moreover, we investigate the semimartingale structure of the infinite dimensional reflecting Ornstein-Uhlenbeck process for open sets of the form $\mathscr O=\{x\in E\, :\, G(x)<0\}$, where $ G$ is some Borel function on $E$.

Quantization of Gaussian measures with Rényi-α-entropy constraints

We consider the optimal quantization problem with Rényi-$α$-entropy constraints for centered Gaussian measures on a separable Banach space. For $α= \infty$ we can compute the optimal quantization error by a moment on a ball. For $α\in {}]1,\infty]$ and large entropy bound we derive sharp asymptotics for the optimal quantization error in terms of the small ball probability of the Gaussian measure. We apply our results to several classes of Gaussian measures. The asymptotical order of the optimal quantization error for $α> 1$ is different from the well-known cases $α= 0$ and $α= 1$.

Monochromatic Subgraphs in Randomly Colored Graphons

Let $T(H, G_n)$ be the number of monochromatic copies of a fixed connected graph $H$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we give a complete characterization of the limiting distribution of $T(H, G_n)$, when $\{G_n\}_{n \geq 1}$ is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, $T(H, G_n)$ either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, $T(H, G_n…

The maximum of log-correlated Gaussian fields in random environments

We study the distribution of the maximum of a large class of Gaussian fields indexed by a box $V_N\subset Z^d$ and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding, Roy and Zeitouni (Annals Probab. (45) 2017, 3886-3928), we show that asymptotically, the centered maximum of the field has a randomly-shifted Gumbel distribution. We prove that the two dimensional Gaussian free field on a super-critical bond percolation cluster with $p$ close enough to $1$, as well as the Gaussian free field in i.i.d. bou…

More on the convergence of Gaussian convex hulls

A "law of large numbers" for consecutive convex hulls for weakly dependent Gaussian sequences $\{X_n\}$, having the same marginal distribution, is extended to the case when the sequence $\{X_n\}$ has a weak limit. Let $\mathbb{B}$ be a separable Banach space with a conjugate space $\mathbb{B}^\ast$. Let $\{X_n\}$ be a centered $\mathbb{B}$-valued Gaussian sequence satisfying two conditions: 1) $X_n \Rightarrow X\;\;$ and 2) For every $x^* \in \mathbb{B}^\ast$ $$ \lim_ {n,m, |n-m|\rightarrow \infty}E\langle X_n, x^*\rangle \langle X_m, x^*\rangle\;\; = \;\;0. $$ Then with probability 1 the norm…

Sobolev spaces with respect to a weighted Gaussian measures in infinite dimensions

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,μ)$ and $\log w\in W^{1,t}(X,μ)$ for some $s>1$ and $t>s'$. In the present paper we introduce and study Sobolev spaces with respect to the weighted Gaussian measure $ν:=wμ$. We obtain results regarding the divergence operator (i.e. the adjoint in $L^2$ of the gradient operator along the Cameron--Martin space) and the trace of Sobolev functions on hypersurfaces $\{x\in X\,|\, G(x) = 0\}$, where $G$ is a suitable version of a Sobolev …

On Besov regularity of Brownian motions in infinite dimensions

We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form $\E \sup_{n\geq 1}\|ξ_n\|$, where the $ξ_n$ are independent centered Gaussian random variables with va…

Covering a compact space by fixed-radius or growing random balls

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a "weak concentration" bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the f…

分层格子上次临界DG模型最大值的极限定律

A limit law for the maximum of subcritical DG-model on a hierarchical lattice

研究分层格子上次临界DG模型的极值性质，通过重正化群分析证明其中心化最大值依分布收敛至随机平移Gumbel律的离散变体，且极值过程收敛至装饰泊松点过程。

渗流超立方图中独立集的高斯到对数正态转变

Gaussian to log-normal transition for independent sets in a percolated hypercube

该研究针对超立方图键渗流子图上的硬核模型配分函数，结合概率框架与团簇展开，证明其涨落在p>2/3时呈高斯分布、p=2/3时呈双对数正态和分布，揭示了独立集的相变几何特征。