相关论文: On Hoeffding's inequalities

Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that $M_n$ is singular is at most $(3/4…

For independent $X$ and $Y$ in the inequality $P(X\leq Y+\mu)$, we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds…

In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the…

The paper is a continuation of our paper [12,2], and it studies functional inequalities for non-local Dirichlet forms with finite range jumps or large jumps. Let $\alpha\in(0,2)$ and $\mu_V(dx)=C_Ve^{-V(x)}\,dx$ be a probability measure. We…

New lower bounds on the total variation distance between the distribution of a sum of independent Bernoulli random variables and the Poisson random variable (with the same mean) are derived via the Chen-Stein method. The new bounds rely on…

We prove that the Birkhoff sum S(n)/n = (1/n) sum_(k=1)^(n-1) g(k A) with g(x) = cot(Pi x) and golden ratio A converges in the sense that the sequence of functions s(x) = S([ x q(2n)])/q(2n) with Fibonacci numbers q(n) converges to a self…

The T product operation between two three order tensors was invented around 2011 and it arises from many applications, such as signal processing, image feature extraction, machine learning, computer vision, and the multiview clustering…

Under the Ornstein-Uhlenbeck semigroup $\{U_t\}$, any non-negative measurable $f : \mathbb R^n \to \mathbb R_+$ exhibits a uniform tail bound better than that implied by Markov's inequality and conservation of mass: For every $\alpha \geq…

Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and…

We consider the sums $S_n=\xi_1+\cdots+\xi_n$ of independent identically distributed random variables. We do not assume that the $\xi$'s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the…

We prove the Simons-Johnson theorem for the sums $S_n$ of $m$-dependent random variables, with exponential weights and limiting compound Poisson distribution $\CP(s,\lambda)$. More precisely, we give sufficient conditions for…

The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Pe\~{n}a, Pinelis and van de Geer. Moreover,…

In Scott (2002) and Congdon (2006), a new method is advanced to compute posterior probabilities of models under consideration. It is based solely on MCMC outputs restricted to single models, i.e., it is bypassing reversible jump and other…

This paper considers a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win…

We present a new concentration of measure inequality for sums of independent bounded random variables, which we name a split-kl inequality. The inequality is particularly well-suited for ternary random variables, which naturally show up in…

Inspired by the idea of Bernoulli decomposition, we give a simple proof for a generalization of Hal\'asz anti--concentration result about random sum of vectores in $\mathbb{R}^d$. From our results, we can give one upper bound for the…

We revisit and refine known tail inequalities and confidence bounds for the hypergeometric distribution, i.e., for the setting where we sample without replacement from a fixed population with binary values or properties. The results are…

The prime counting function inequality $\pi(x+y) < \pi(x)+\pi(y)$, which is known as Hardy-Littlewood conjecture, has been established for a variety of cases such as $ \delta x \leq y \leq x$, where $0< \delta \leq 1$, and $x \leq y\leq x…

A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes…

About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings…