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Grzegorz Serafin

Publications and source records attributed to Grzegorz Serafin.

At least 19 recordsLinked to original sources
arXiv2025

Normal approximation for number of edges in random intersection graphs

The random intersection graph model $\mathcal G(n,m,p)$ is considered. Due to substantial edge dependencies, studying even fundamental statistics such as the subgraph count is significantly more challenging than in the classical binomial model $\mathcal G(n,p)$. First, we establish normal approximation bound in both the Wasserstein and the Kolmogorov distances for a class of local statistics on $\mathcal G(n,m,p)$. Next, we apply these results to derive such bounds for the standardised number of edges, and determine the necessary and sufficient conditions for its asymptotic normality. We develop a new method that provides a combinatorial interpretation and facilitates the estimation of analytical expressions related to general distance bounds. In particular, this allows us to control the behaviour of central moments of subgraph existence indicators. The presented method can also be extended to count copies of subgraphs larger than a single edge.

math.CO↗
arXiv2024

Explicit bounds for Bell numbers and their ratios

In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main results correspond to two asymptotic forms expressed by means of the Lambert $W$ function. As an application, some straightforward elementary bounds are derived. Additionally, an absolute convergence rate of the ratio of the consecutive Bell numbers is derived. The main challenge was to obtain satisfactory constants, as the Bell numbers grow rapidly, while the convergence rates are rather slow.

math.NT↗
arXiv2024

Normal approximation for subgraph count in random hypergraphs

A non-uniform and inhomogeneous random hypergraph model is considered, which is a straightforward extension of the celebrated binomial random graph model $\mathbb G(n, p)$. We establish necessary and sufficient conditions for small hypergraph count to be asymptotically normal, and complement them with convergence rate in both the Wasserstein and Kolmogorov distances. Next we narrow our attention to the homogeneous model and relate the obtained results to the fourth moment phenomenon. Additionally, a short proof of necessity of aforementioned conditions is presented, which seems to be absent in the literature even in the context of the model $\mathbb G(n, p)$.

math.PR↗
arXiv2022

Fractional Burgers equation with singular initial condition

We consider the fractional Burgers equation $ Δ^{α/2} u + b\cdot \nabla (u|u|^{(α-1)/β})$ on ${\mathbf R}^d$, $d\geq2$, with {$α\in (1,2)$ and} $β>1$ and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any $L^p({\mathbf R}^d)$, $1\leq p\leq\infty$. Next, we apply the general results to the initial condition $u_0(x)=M|x|^{-β}$, $1<β<d$, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.

math.AP↗
arXiv2021

Backward Touchard congruence

The celebrated Touchard congruence states that $B_{n+p}B_n+B_{n+1}$ modulo $p$, where $p$ is a prime number and $B_n$ denotes the Bell number. In this paper we study divisibility properties of $B_{n-p}$ and their generalizations involving higher powers of $p$ as well as the $r$-Bell numbers. In particular, we show a closely relation of the considered problem to the Sun-Zagier congruence, which is additionally improved by deriving \mbox{a new} relation between $r$-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo $p$.

math.NT↗
arXiv2021

Laplace Dirichlet heat kernels in convex domains

We provide general lower and upper bounds for Laplace Dirichlet heat kernel of convex $\mathcal C^{1,1}$ domains. The obtained estimates precisely describe the exponential behaviour of the kernels, which has been known only in a few special cases so far. Furthermore, we characterize a class of sets for which the estimates are sharp, i.e. the upper and lower bounds coincide up to a multiplicative constant. In particular, this includes sets of the form $\{x\in \R^n: x_n>a|(x_1,...,x_{n-1})|^p\}$ where $p\geq2$, $n\geq2 $ and $a>0$.

math.AP↗
arXiv2020

Berry-Esseen bounds for functionals of independent random variables

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on chaos expansions methods. Our results apply to $U$-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate $U$-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weighted $U$-statistics, quadratic forms, and random subgraph weighing. In the case of quadratic forms, our results recover and improve the bounds available in the literature, and apply to matrices with non-empty diagonals.

math.PR↗
arXiv2020

Normal approximation for generalized U-statistics and weighted random graphs

We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erdős-Rényi random graph, extending the graph counting results of [1] to the setting of graph weighting. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences.

math.PR↗
arXiv2020

Feeling boundary by Brownian motion in a ball

We provide short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. The boundary behaviour is precisely described. Presented results may be considered as a complement or a generalization of the famous "principle of not feeling the boundary" in case of a ball.

math.PR↗
arXiv2019

On potential theory of hyperbolic Brownian motion with drift

Consider the $λ$-Green function and the $λ$-Poisson kernel of a Lipschitz domain $U\subset \mathbb H^n=\left\{x\in\mathbb R^n:x_n>0\right\}$ for hyperbolic Brownian motion with drift. We provide several relationships that facilitate studying those objects and explain somehow theirs nature. As an application, we yield uniform estimates in case of sets of the form $S_{a,b}=\{x\in\mathbb H^n:x_n>a, x_1\in(0,b)\}$, $a,b>0$, $a,b>0$, which covers and extends existing results of that kind.

math.PR↗
arXiv2018

Uniform pointwise asymptotics of solutions to quasi-geostrophic equation

We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in $L^{2/(α-1)}(\mathbb{R}^2)$. Furthermore, we give upper bound of similar type for any derivative of the solutions. Initial data in $L^{p}(\mathbb{R}^2)$, $p>2/(α-1)$, are also discussed.

math.AP↗
arXiv2018

Normal approximation for sums of discrete $U$-statistics - application to Kolmogorov bounds in random subgraph counting

We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraphs counts in the Erd{\H o}s-Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering and improving recent results derived for triangles as well as results using the Wasserstein distance.

math.PR↗
arXiv2018

Stein approximation for functionals of independent random sequences

We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and functionals of both continuous and discrete independent random variables. For random variables admitting a continuous density, it recovers classical distance bounds based on absolute third moments, with better and explicit constants. We also apply this method to multiple stochastic integrals that can be used to represent U-statistics, and include linear and quadratic functionals as particular cases.

math.PR↗
arXiv2017

Dirichlet heat kernel for the Laplacian in a ball

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel $k_1(t,x,y)$ in terms of its global two-sided sharp estimates.

math.AP↗
arXiv2016

Lévy systems and moment formulas for mixed Poisson integrals

We propose Mecke-Palm formulas for multiple integrals with respect to a Poisson random measure interlaced with its intensity measure. We apply such formulas to multiple mixed Lévy systems of Lévy processes and obtain moment formulas for products of interlaced multiple Poisson integrals.

math.PR↗
arXiv2015

Pointwise estimates for solutions of fractal Burgers equation

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $α\in(1,2)$, $b\in\mathbb R^d$ for $q = (α-1)/d$ and $u_0 \in L^1(\mathbb R^d)$. We consider also $q > (α-1)/d$ under additional condition $u_0 \in L^\infty(\mathbb R^d)$. In both cases we assume $u_0\geq0$, which implies that the solution is non-negative. The estimates are given in the terms of the function $P_t u_0$, where $P_t$ is the stable semigroup operator.

math.AP↗
arXiv2015

Exit times densities of Bessel process

We examine the density functions of the first exit times of the Bessel process from the intervals [0,1) and (0,1). First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the n-dimensional Brownian motion from a ball.

math.PR↗