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Péter Kevei

Publications and source records attributed to Péter Kevei.

At least 19 recordsLinked to original sources
arXiv2026

Higher moments of intrinsic volumes of random beta-prime polytopes

We consider beta-prime polytopes, i.e., the convex hulls of iid random points chosen according to beta-prime distributions in $\mathbb{R}^d$. After suitable scaling, beta-prime polytopes converge in distribution to the convex hulls of Poisson point processes with power-law intensity functions. We prove moment convergence for the volume and all intrinsic volumes. Beta-prime polytopes are the push-forwards of spherical random polytopes on the upper open half-sphere of the unit sphere $S^d\subset \mathbb{R}^{d+1}$. We prove convergence of moments of the spherical volume difference of the half-sphere and the spherical random polytopes.

math.MG↗
arXiv2026

Path properties of Lévy driven mixed moving average processes

We derive general sufficient conditions for the existence of càdlàg and continuous modifications of Lévy-driven mixed moving average processes. The conditions are explicit and easy to verify and applied to supOU, well-balanced supOU, trawl, and power-weighted supOU processes. In these examples, the conditions are shown to be close to optimal.

math.PR↗
arXiv2026

Marcinkiewicz--Zygmund-type SLLN for mixed moving average processes

The Marcinkiewicz--Zygmund theorem is a fundamental result in probability theory that establishes rates of convergence in the strong law of large numbers (SLLN). Although numerous extensions have been developed for dependent sequences, many classes of processes, particularly those exhibiting strong dependence, remain unexplored. In this paper, we present a Marcinkiewicz--Zygmund-type SLLN for a class of mixed moving average processes, which form a large and flexible class of stationary infinitely divisible processes. In contrast to the classical case, where moments determine the asymptotic behavior, the present setting additionally involves key objects that characterize both dependence and marginal distributions.

math.PR↗
arXiv2025

Generalized Rényi statistics

In Rényi's representation for exponential order statistics, we replace the iid exponential sequence with any iid sequence, and call the resulting order statistic generalized Rényi statistic. We prove that by randomly reordering the variables in the generalized Rényi statistic, we obtain in the limit a sequence of iid exponentials. This result allows us to propose a new model for heavy-tailed data. Although the new model is very close to the classical iid framework, we establish that the Hill estimator is weakly consistent and asymptotically normal without any further assumptions on the underlying distribution or on the number of upper order statistics used in the estimator.

math.ST↗
arXiv2023

Branching model with state dependent offspring distribution for Chlamydia spread

Chlamydiae are bacteria with an interesting unusual developmental cycle. A single bacterium in its infectious form (elementary body, EB) enters the host cell, where it converts into its dividing form (reticulate body, RB), and divides by binary fission. Since only the EB form is infectious, before the host cell dies, RBs start to convert into EBs. After the host cell dies RBs do not survive. We model the population growth by a 2-type discrete-time branching process, where the probability of duplication depends on the state. Maximizing the EB production leads to a stochastic optimization problem. Simulation study shows that our novel model is able to reproduce the main features of the development of the population.

q-bio.PE↗
arXiv2022

Nearly critical Galton--Watson processes

We investigate Galton--Watson processes in varying environment, for which $\bar f_n \uparrow 1$ and $\sum_{n=1}^\infty (1-\bar f_n) = \infty$, where $\bar f_n$ stands for the offspring mean in generation $n$. Since the process dies out almost surely, to obtain nontrivial limit we consider two scenarios: conditioning on non-extinction, or adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).

math.PR↗
arXiv2022

A landscape of peaks: The intermittency islands of the stochastic heat equation with Lévy noise

We show that the spatial profile of the solution to the stochastic heat equation features multiple layers of intermittency islands if the driving noise is non-Gaussian. On the one hand, as expected, if the noise is sufficiently heavy-tailed, the largest peaks of the solution will be taller under multiplicative than under additive noise. On the other hand, surprisingly, as soon as the noise has a finite moment of order $\frac2d$, where $d$ is the spatial dimension, the largest peaks will be of the same order for both additive and multiplicative noise, which is in sharp contrast to the behavior of the solution under Gaussian noise. However, in this case, a closer inspection reveals a second layer of peaks, beneath the largest peaks, that is exclusive to multiplicative noise and that can be observed by sampling the solution on the lattice. Finally, we compute the macroscopic Hausdorff and Minkowski dimensions of the intermittency islands of the solution. Under both additive and multiplicative noise, if it is not too heavy-tailed, the largest peaks will be self-similar in terms of their large-scale multifractal behavior. But under multiplicative noise, this type of self-similarity is not present in the peaks observed on the lattice.

math.PR↗
arXiv2022

Extremes of the stochastic heat equation with additive Lévy noise

We analyze the spatial asymptotic properties of the solution to the stochastic heat equation driven by an additive Lévy space-time white noise. For fixed time $t > 0$ and space $x \in \mathbb{R}^d$ we determine the exact tail behavior of the solution both for light-tailed and for heavy-tailed Lévy jump measures. Based on these asymptotics we determine for any fixed time $t> 0$ the almost-sure growth rate of the solution as $|x| \to \infty$.

math.PR↗
arXiv2021

On random disc-polygons in a disc-polygon

We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of Rényi and Sulanke (1964) about random polygons in convex polygons.

math.MG↗
arXiv2021

Stochastic modeling of in vitro bactericidal potency

We provide a Galton--Watson model for the growth of a bacterial population in the presence of antibiotics. We assume that bacterial cells either die or duplicate, and the corresponding probabilities depend on the concentration of the antibiotic. Assuming that the mean offspring number is given by $m(c) = 2 / (1 + αc^β)$ for some $α, β$, where $c$ stands for the antibiotic concentration we obtain weakly consistent, asymptotically normal estimator both for $(α, β)$ and for the minimal inhibitory concentration (MIC), a relevant parameter in pharmacology. We apply our method to real data, where \emph{Chlamydia trachomatis} bacteria was treated by azithromycin and ciprofloxacin. For the measurements of \emph{Chlamydia} growth quantitative PCR technique was used. The 2-parameter model fits remarkably well to the biological data.

stat.AP↗
arXiv2020

Statistical inference of subcritical strongly stationary Galton--Watson processes with regularly varying immigration

We describe the asymptotic behavior of the conditional least squares estimator of the offspring mean for subcritical strongly stationary Galton--Watson processes with regularly varying immigration with tail index $α\in (1,2)$. The limit law is the ratio of two dependent stable random variables with indices $α/2$ and $2α/3$, respectively, and it has a continuously differentiable density function. We use point process technique in the proofs.

math.ST↗
arXiv2019

The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise

We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a Lévy space-time white noise. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. More precisely, if we normalize the solution by an increasing nonnegative function, we either obtain convergence to $0$, or the limit superior and/or inferior will be infinite. A detailed analysis of the jumps further reveals that the strong law of large numbers can be recovered on discrete sequences of time points increasing to infinity. This leads to a necessary and sufficient condition that depends on the Lévy measure of the noise and the growth and concentration properties of the sequence at the same time. Finally, we show that our results generalize to the stochastic heat equation with a multiplicative nonlinearity that is bounded away from zero and infinity.

math.PR↗
arXiv2018

Intermittency for the stochastic heat equation with Lévy noise

We investigate the moment asymptotics of the solution to the stochastic heat equation driven by a $(d+1)$-dimensional Lévy space--time white noise. Unlike the case of Gaussian noise, the solution typically has no finite moments of order $1+2/d$ or higher. Intermittency of order $p$, that is, the exponential growth of the $p$th moment as time tends to infinity, is established in dimension $d=1$ for all values $p\in(1,3)$, and in higher dimensions for some $p\in(1,1+2/d)$. The proof relies on a new moment lower bound for stochastic integrals against compensated Poisson measures. The behavior of the intermittency exponents when $p\to 1+2/d$ further indicates that intermittency in the presence of jumps is much stronger than in equations with Gaussian noise. The effect of other parameters like the diffusion constant or the noise intensity on intermittency will also be analyzed in detail.

math.PR↗
arXiv2016

Bahadur--Kiefer Representations for Time Dependent Quantile Processes

We define a time dependent empirical process based on $n$ independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur--Kiefer representations.

math.PR↗
arXiv2016

Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum

In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determine the limit distribution of the St. Petersburg sum conditioning on its maximum, and we analyze how the limit depends on the value of the maximum. As an application, we obtain an infinite sum representation of the distribution function of the possible semistable limits. In the representation, each term corresponds to a given maximum, in particular this result explains that the semistable behavior is caused by the typical values of the maximum.

math.PR↗
arXiv2015

Tail probabilities of St. Petersburg sums, trimmed sums, and their limit

We provide exact asymptotics for the tail probabilities $\mathbb{P} \{S_{n,r} > x\}$ as $x \to \infty$, for fix $n$, where $S_{n,r}$ is the $r$-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the $r$-trimmed limits.

math.PR↗