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The full moments expansion of the joint probability distribution of an isotropic random field, its gradient and invariants of the Hessian is presented in 2 and 3D. It allows for explicit expression for the Euler characteristic in ND and…

Let $X_1, \ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $\mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large…

The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying…

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the…

We investigate extreme value theory for physical systems with a global conservation law which describe renewal processes, mass transport models and long-range interacting spin models. As shown previously, a special feature is that the…

We obtain the analogue of the classical result by Erd\"os and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central…

We construct a new sufficient conditions for boundedness or continuity of arbitrary random fields relying on the so-called partition scheme, alike in the classical majorizing measure method. We deduce also the used in the practice…

We introduce the problem of estimation of the parameters of a dynamically selected population in an infinite sequence of random variables and provide its application in the statistical inference based on record values from a non-stationary…

We develop a simple and unified approach to investigate several aspects of the cluster statistics of random expansive (multi-)sets. In particular, we determine the limiting distribution of the size of the smallest and largest clusters, we…

The task of the binary classification problem is to determine which of two distributions has generated a length-$n$ test sequence. The two distributions are unknown; two training sequences of length $N$, one from each distribution, are…

We prove large (and moderate) deviations for a class of linear combinations of spacings generated by i.i.d. exponentially distributed random variables. We allow a wide class of coefficients which can be expressed in terms of continuous…

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). We show that if there exists a sequence of positive real numbers $(t_n)_{n=1}^{\infty}$ converging to 0 such that $$\lim_{N \rightarrow \infty}{…

For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…

We consider the estimation of the mixing distribution of a normal distribution where both the shift and scale are unobserved random variables. We argue that in general, the model is not identifiable. We give an elegant non-constructive…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process…

The likelihood function is central to both frequentist and Bayesian formulations of parametric statistical inference, and large-sample approximations to the sampling distributions of estimators and test statistics, and to posterior…

This expository note describes how to apply the method of maximum likelihood to estimate the parameters of the ``$q$-exponential'' distributions introduced by Tsallis and collaborators. It also describes the relationship of these…

In most data-scientific approaches, the principle of Maximum Entropy (MaxEnt) is used to a posteriori justify some parametric model which has been already chosen based on experience, prior knowledge or computational simplicity. In a…

Let $X_{i,n},n\in \mathbb{N},1\leq i\leq n$, be a triangular array of independent $\mathbb{R}^d$-valued Gaussian random vectors with correlation matrices $\Sigma_{i,n}$. We give necessary conditions under which the row-wise maxima converge…