Related papers: Explicit rates of approximation in the CLT for qua…

Let $X_1, \ldots , X_n$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\|X_1\| \le \beta$. Then, we show that $\frac{1}{\sqrt{n}}(X_1 + \ldots + X_n)$ converges to a Gaussian in quadratic transportation (also known as "Kantorovich" or…

This paper is concerned with normal approximation under relaxed moment conditions using Stein's method. We obtain the explicit rates of convergence in the central limit theorem for (i) nonlinear statistics with finite absolute moment of…

We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in a sequence $X = (X_{i})$ of independent and identically distributed random variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a…

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_1,\dots,x_d)=1$ over the real…

Let $X_1,\dots, X_n$ be independent and identically distributed random vectors in $\mathbb{R}^d$. Suppose $\mathbb{E} X_1=0$, $\mathrm{Cov}(X_1)=I_d$, where $I_d$ is the $d\times d$ identity matrix. Suppose further that there exist positive…

In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,\dots,X_n$ over the class of rectangles in the case when the covariance…

In computational and applied statistics, it is of great interest to get fast and accurate calculation for the distributions of the quadratic forms of Gaussian random variables. This paper presents a novel approximation strategy that…

In this paper we characterize all distributional limits of the random quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u, v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal and…

We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It…

Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals…

In this paper, we use the dimensional reduction technique to study the central limit theory (CLT) random quadratic forms based on sample means and sample covariance matrices. Specifically, we use a matrix denoted by $U_{p\times q}$, to map…

We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} \theta_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and…

Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in $D[0, 1]$ for the partial sum processes $\{S_{[nt]}, 0\le t\le 1\}$, where…

We compute the exact rates of convergence in total variation associated with the 'fourth moment theorem' by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit…

In this note, we assess the accuracy of CLT-based approximations for the volume of intersection of the $d$-dimensional cube $[-1,1]^d$ and an $L_q$-ball centred at the origin; this is clearly equivalent to approximating the distribution of…

For $k,m,n\in \mathbb{N}$, we consider $n^k\times n^k$ random matrices of the form $$ \mathcal{M}_{n,m,k}(\mathbf{y})=\sum_{\alpha=1}^m\tau_\alpha {Y_\alpha}Y_\alpha^T,\quad…

Ratios of quadratic forms in correlated normal variables which introduce noncentrality into the quadratic forms are considered. The denominator is assumed to be positive (with probability 1). Various serial correlation estimates such as…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

This paper focuses on inhomogeneous quadratic tests, which involve the sum of a dependent non-central chi-square with a Gaussian random variable. Unfortunately, no closed-form expression is available for the statistical distribution of the…

This note presents sharp inequalities for deviation probability of a general quadratic form of a random vector \(\xiv\) with finite exponential moments. The obtained deviation bounds are similar to the case of a Gaussian random vector. The…