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We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of…

We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order…

We enable a theory of intrinsic asymptotic expansions for the steady state solutions of the full Navier-Stokes equations. Such a theory was first developed in Foias et al (2024 Commun. Pure Appl. Anal. 23, 269-303) for Galerkin…

We obtain the law of large numbers (LLN) and the central limit theorem (CLT) for weakly dependent non-stationary arrays of random fields with asymptotically unbounded moments. The weak dependence condition for arrays of random fields is…

This paper is concerned with the Stein's method associated with a (possibly) asymmetric $\alpha$-stable distribution $Z$, in dimension one. More precisely, its goal is twofold. In the first part, we exhibit a genuine bound for the…

We consider the problem of estimating smooth integrated functionals of a monotone nonincreasing density $f$ on $[0,\infty)$ using the nonparametric maximum likelihood based plug-in estimator. We find the exact asymptotic distribution of…

We consider 1-dimensional location estimation, where we estimate a parameter $\lambda$ from $n$ samples $\lambda + \eta_i$, with each $\eta_i$ drawn i.i.d. from a known distribution $f$. For fixed $f$ the maximum-likelihood estimate (MLE)…

Understanding rare events is critical across domains ranging from signal processing to reliability and structural safety, extreme-weather forecasting, and insurance. The analysis of rare events is a computationally challenging problem,…

We present, in a unified way, a Stein methodology for infinitely divisible laws (without Gaussian component) having finite first moment. Based on a correlation representation, we obtain a characterizing non-local Stein operator which boils…

Given a probability measure on a finitely generated group, the local limit problem consists in finding asymptotics of $p_n(e,e)$, the probability that the random walk at time $n$ is at the origin. We give the classification of all possible…

For a numerical sequence ${a_n}$ satisfying broad assumptions on its "behaviour on average" and a random walk $S_n=\xi_1 +...+\xi_n$ with i.i.d. jumps $\xi_j$ with positive mean $\mu$, we establish the asymptotic behaviour of the sums…

Let $\nu\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in \mathbb{N}$, let $(X_n^{p})_{n\geq1}$ be i.i.d. $\mathbb{R}^p$-valued random variables with radially symmetric distributions and radial distribution…

This study considers the unconditional smooth R\'{e}nyi entropy, the smooth conditional R\'{e}nyi entropy proposed by Kuzuoka [\emph{IEEE Trans.\ Inf.\ Theory}, vol.~66, no.~3, pp.~1674--1690, 2020], and a new quantity which we term the…

We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope $P$ as their degree grow. We consider a large class of probability distributions including the ones…

Providing theoretical guarantees for parameter estimation in exponential random graph models is a largely open problem. While maximum likelihood estimation has theoretical guarantees in principle, verifying the assumptions for these…

For a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ and $B(t)= \int_{0}^{t} d\tau \delta(X(\tau)-L) $ at…

In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count…

We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in…

We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is determined by the Lorentz-critical regularity \( s_L = 2 \), and both local…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…