Related papers: Large Deviations Application to Billingsley's Exam…
This paper provides a quantitative version of de Finetti law of large numbers. Given an infinite sequence $\{X_n\}_{n \geq 1}$ of exchangeable Bernoulli variables, it is well-known that $\frac{1}{n} \sum_{i = 1}^n X_i…
We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our…
An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(\lambda_1,\cdots,\lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form…
Let $F_n$ denote the distribution function of the normalized sum $Z_n = (X_1 + \dots + X_n)/\sigma\sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal…
The paper deals with the fast-slow motions setups in the discrete time $X^\epsilon((n+1)\epsilon)=X^\epsilon(n\epsilon)+\epsilon B(X^\epsilon(n\epsilon),\xi(n))$, $n=0,1,...,[T/\epsilon]$ and the continuous time $\frac…
Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $P(\max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$…
This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation $D(f)$ by a weighted relative Fisher information of $f$ with…
From a simple path integral involving a variable volatility in the velocity differences, we obtain velocity probability density functions with exponential tails, resembling those observed in fully developed turbulence. The model yields…
This paper addresses the statistical problem of estimating the infinite-norm deviation from the empirical mean to the distribution mean for high-dimensional distributions on $\{0,1\}^d$, potentially with $d=\infty$. Unlike traditional…
Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an…
Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…
We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is $$ \varepsilon_{n}(f):={\mathbb{E}}\Big(f\Big(\frac 1{\sqrt…
Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers…
In group sequential analysis, data is collected and analyzed in batches until pre-defined stopping criteria are met. Inference in the parametric setup typically relies on the limiting asymptotic multivariate normality of the repeatedly…
Motivated by the weak limit of the Kolmogorov-Smirnov test statistics, in this contribution, we concern the asymptotics of \begin{align*} \mathbb{P}\left\{\sup_{\boldsymbol{x}\in [0,1]^n}\left(W(\boldsymbol{x})\Big|…
The standard Large Deviation Theory (LDT) is mathematically illustrated by the Boltzmann-Gibbs factor which describes the thermal equilibrium of short-range-interacting many-body Hamiltonian systems, the velocity distribution of which is…
An explicit bound is given for the Kolmogorov distance between a mixture of normal distributions and a normal distribution with properly chosen parameter values. A random variable X has a mixture of normal distributions if its conditional…
We show that the moment generating function of the Kullback-Leibler divergence (relative entropy) between the empirical distribution of $n$ independent samples from a distribution $P$ over a finite alphabet of size $k$ (i.e. a multinomial…
In terms of the Dirac representation of sample mean and the weak convergence of empirical distributions that holds almost surely, we construct a new proof for a strong law of large numbers of Kolmogorov's type with i.i.d. random variables…
We develop a variant of Stein's method of comparison of generators to bound the Kolmogorov, total variation, and Wasserstein-1 distances between distributions on the real line. Our discrepancy is expressed in terms of the ratio of reverse…