Related papers: First-order convergence for $321$-avoiding permuta…

We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance…

We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution…

We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild…

Nourdin et al. [9] established the following universality result: if a sequence of off-diagonal homogeneous polynomial forms in i.i.d. standard normal random variables converges in distribution to a normal, then the convergence also holds…

Nesetril and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no…

The Erd\H os-R\'enyi law states that given a sequence $\{X_j\}_{j=1}^\infty$ of i.i.d.~($p$) coin-tosses, the longest run $L_n$ of heads in the first $n$ coin tosses approaches $\log_{1/p}n$ almost surely. In this paper we explore a…

This note proves a law of large numbers for predicting several steps ahead, which, in the case of uniformly bounded random variables, generalizes the standard law of large numbers for martingales; the standard law of large numbers…

We prove convergence rates of monotone schemes for conservation laws for H\"older continuous initial data with unbounded total variation, provided that the H\"older exponent of the initial data is greater than $1/2$. For strictly…

Max stable laws are limit laws of linearly normalized partial maxima of independent identically distributed random variables. Saeb (2014) proves that the Renyi entropy of order b (b > 1) of linear normalized maximum of iid random variables…

In a previous paper, we showed that $3\bar{5}241$-avoiding permutations are counted by the unique sequence that starts with a 1 and shifts left under the self-composition transform. The proof uses a complicated bijection. Here we give a…

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that…

We apply the Inclusion-Exclusion Principle to a unique pair of prime number subsequences to determine whether these subsequences form a small set or a large set and thus whether the infinite sum of the inverse of their terms converges or…

We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the…

The number of 123-avoiding permutation on $\{1,2,\ldots,n\}$ with a fixed leading terms is counted by the ballot numbers. The same holds for $132$-avoiding permutations. These results were proved by Miner and Pak using the…

Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under…

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained…

The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a…

The main result is the following Theorem: Let p=p(n) be such that p(n) in [0,1] for all n and either p(n)<< n^{-1} or for some positive integer k, n^{-1/k}<< p(n)<< n^{-1/(k+1)} or for all epsilon >0, n^{- epsilon}<< p(n) and n^{-…

We construct a refined bijection $\phi$ between alternating permutations and 0-1-2 increasing trees with degree at most 2. It satisfies that the first element of alternating permutation $\pi$ is equal to the first vertex in $\phi(\pi)$ in…

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability…