Related papers: Bounds for the loss probability in large loss queu…
The probability distribution of the maximum $M_t$ of a single resetting Brownian motion (RBM) of duration $t$ and resetting rate $r$, properly centred and scaled, is known to converge to the standard Gumbel distribution of the classical…
In time series analysis there is an apparent dichotomy between time and frequency domain methods. The aim of this paper is to draw connections between frequency and time domain methods. Our focus will be on reconciling the Gaussian…
There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general…
We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the…
For Gaussian random fields with values in $\mathbb{R}^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For…
Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $N(n,m)$ denote…
Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this…
Given a graph $G$, we consider a model for a random cover of $G$ by taking two parallel copies of $G$ and crossing every pair of parallel edges randomly with probability $q$ independently of each other. The resulting graph $G_q$, is a…
Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and width constraints yields the Gaussian distribution. Also, the basic solutions of the standard Fokker-Planck (FP) equation (related to the Langevin equation with…
In "Recognizing the Maximum of a Sequence", Gilbert and Mosteller analyze a full information game where n measurements from an uniform distribution are drawn and a player (knowing n) must decide at each draw whether or not to choose that…
Under the assumption that the distribution of a nonnegative random variable $X$ admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to…
This paper deals with optimally-robust parameter estimation in generalized Pareto distributions (GPDs). These arise naturally in many situations where one is interested in the behavior of extreme events as motivated by the…
Paired comparison data considered in this paper originate from the comparison of a large number N of individuals in couples. The dataset is a collection of results of contests between two individuals when each of them has faced n opponents,…
We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\ge0)$ and its empirical spectral measure process $\mu^{(n)}=(\mu_{t}^{(n)};t\ge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an…
This paper studies binary classification problem associated with a family of loss functions called large-margin unified machines (LUM), which offers a natural bridge between distribution-based likelihood approaches and margin-based…
This paper extends the standard chaining technique to prove excess risk upper bounds for empirical risk minimization with random design settings even if the magnitude of the noise and the estimates is unbounded. The bound applies to many…
We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and…
We introduce the notion of a Young generating function for a probability measure on integer partitions. We use this object to characterize probability distributions over integer partitions satisfying a law of large numbers and those that…
It is common to model random errors in a classical measurement by the normal (Gaussian) distribution, because of the central limit theorem. In the quantum theory, the analogous hypothesis is that the matrix elements of the error in an…
We consider the problem of calculating learning curves (i.e., average generalization performance) of Gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue…