Related papers: Malliavin Calculus and Self Normalized Sums
We show that the distribution of self-normalized sums of free self-adjoint random variables converges weakly to Wigner's semicircle law under appropriate conditions and estimate the rate of convergence in terms of the Kolmogorov distance.…
We derive a Gaussian Central Limit Theorem for the sample quantiles based on locally dependent random variables with explicit convergence rate. Our approach is based on converting the problem to a sum of indicator random variables, applying…
We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random…
This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations…
We construct a non - improved exponential bounds for distribution of normed sums of i.,i.d. random variables with random numbers of summand.
We consider Malliavin calculus based on the It\^o chaos decomposition of square integrable random variables on the L\'evy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and…
We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cram\'er type large deviations for moderate $x$'s, and are of exponential decay rate as de la Pe\~na's inequality when…
A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of…
We give a simple inequality for the sum of independent bounded random variables. This inequality improves on the celebrated result of Hoeffding in a special case. It is optimal in the limit where the sum tends to a Poisson random variable.
A gem of classical probability, the Berry-Esseen theorem provides a non-asymptotic form of the central limit theorem. This note gives a friendly and intuitive exposition of the classical Fourier-analytic proof of Esseen's smoothing…
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…
We give a nonuniform Berry-Esseen bound for self-normalized martingales, which bridges the gap between the result of Haeusler (1988) and Fan and Shao (2018). The bound coincides with the nonuniform Berry-Esseen bound of Haeusler and Joos…
Let $\mu$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the…
We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that…
Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1/2}$ in the uniform metric, in other words, when…
We obtain Berry-Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $\delta \in (2,\infty]$ using a Fourier transform approach. Our bounds improve the state-of-the-art in the…
Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order $n^{-1/2}$ for studentized…
We prove abstract bounds on the Wasserstein and Kolmogorov distances between non-randomly centered random sums of real i.i.d. random variables with a finite third moment and the standard normal distribution. Except for the case of mean zero…
We extend the main theorem of arXiv:1301.6911 about the fluctuations in the Curie-Weiss model of SOC. We present a short proof using the Hubbard-Stratonovich transformation with the self-normalized sum of the random variables.
An analogue of the Berry-Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n^{-1/2}, the same as in the classical case.…