Related papers: Tail Bounds on the Spectral Norm of Sub-Exponentia…

We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to the square root of the spectral radius of the…

The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several…

Exponential tail bounds for sums play an important role in statistics, but the example of the $t$-statistic shows that the exponential tail decay may be lost when population parameters need to be estimated from the data. However, it turns…

Sums of independent, bounded random variables concentrate around their expectation approximately as well a Gaussian of the same variance. Well known results of this form include the Bernstein, Hoeffding, and Chernoff inequalities and many…

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the…

We derive the mean eigenvalue density for symmetric Gaussian random N x N matrices in the limit of large N, with a constraint implying that the row sum of matrix elements should vanish. The result is shown to be equivalent to a result found…

We prove an exponential probability tail inequality for positive semidefinite quadratic forms in a subgaussian random vector. The bound is analogous to one that holds when the vector has independent Gaussian entries.

We prove that the tail probabilities of sums of independent uniform random variables, up to a multiplicative constant, are dominated by the Gaussian tail with matching variance and find the sharp constant for such stochastic domination.

We show bounds on tail probabilities for quadratic forms in sub-gaussian non-necessarily independent random variables. Our main tool will be estimates of the Luxemburg norms of such forms. This will allow us to formulate the above-mentioned…

An explicit upper bound on the tail probabilities for the normalized Rademacher sums is given. This bound, which is best possible in a certain sense, is asymptotically equivalent to the corresponding tail probability of the standard normal…

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…

This paper deals with symmetric random matrices whose upper diagonal entries are obtained from a linear random field with heavy tailed noise. It is shown that the maximum eigenvalue and the spectral radius of such a random matrix with…

Concentration inequalities form an essential toolkit in the study of high dimensional (HD) statistical methods. Most of the relevant statistics literature in this regard is based on sub-Gaussian or sub-exponential tail assumptions. In this…

We obtain a tail bound for the least non-zero singular value of $A-z$ when $A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a given point $z_0$ in the bulk of the spectrum. The argument relies on a resolvent…

We derive upper bounds on the tail conditional expectation of binomial and Poisson random variables. Those upper bounds are subsequently employed to the problem of obtaining non-asymptotic lower bounds on the probability that the…

We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$,…

In this paper we derive tail bounds on the norms of random submatrices with non-uniformly distributed supports. We apply these results to sparse approximation and conduct an analysis of the average case performance of thresholding,…

In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the…

We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from the work of the second author and Vu.…

Let $(\xi_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |\xi_i|\big/…