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Peter Kevei

Publications and source records attributed to Peter Kevei.

At least 19 recordsLinked to original sources
arXiv2025

Heavy-tailed critical Galton--Watson processes with immigration

Consider a critical Galton--Watson branching process with immigration, where the offspring distribution belongs to the domain of attraction of a $(1 + \alpha)$-stable law with $\alpha \in (0,1)$, and the immigration distribution either (i) has finite mean, or (ii) belongs to the domain of attraction of a $\beta$-stable law with $\beta \in (\alpha, 1)$. We show that the tail of the stationary distribution is regularly varying. We analyze the stationary process, determine its tail process, and establish a stable central limit theorem for the partial sums. The norming sequence is different from the one corresponding to the tail of the stationary law. In particular, the extremal index of the process is $0$.

math.PR↗
arXiv2024

Functional limit theorem for branching processes in nearly degenerate varying environment

We investigate branching processes in nearly degenerate varying environment, where the offspring distribution converges to the degenerate distribution at 1. Such processes die out almost surely, therefore, we condition on non-extinction or add inhomogeneous immigration. Extending our one-dimensional limit results we derive functional limit theorems. In the former case, the limiting process is a time-changed simple birth-and-death process on $(-\infty, 0]$ conditioned on survival at $0$, while in the latter, it is a time-changed stationary continuous time branching process with immigration.

math.PR↗
arXiv2024

Tail behavior and almost sure growth rate of supOU processes

In this paper we consider sample path growth of superpositions of Ornstein--Uhlenbeck type processes (supOU). SupOU processes are stationary infinitely divisible processes defined as integrals with respect to a random measure. They allow marginal distributions and correlations to be modeled independently. Our results show that the almost sure behavior is primarily governed by the tail of the marginal distribution. In particular, we obtain a general integral test for the sample path growth that covers both heavy-tailed and light-tailed scenarios. We also investigate the tail behavior of the marginal distributions in connection with the characteristics of the underlying random measure.

math.PR↗
arXiv2024

Almost sure growth of integrated supOU processes

Superpositions of Ornstein-Uhlenbeck processes allow a flexible dependence structure, including long range dependence for OU-type processes. Their complex asymptotics are governed by three effects: the behavior of the L\'evy measure both at infinity and at zero, and the behavior at zero of the measure governing the dependence. We establish almost sure rates of growth depending on the characteristics of the process and prove a Marcinkiewicz--Zygmund type SLLN for the integrated process.

math.PR↗
arXiv2021

Strong renewal theorem and local limit theorem in the absence of regular variation

We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction a semistable law with index $α\in (1/2,1]$. In the process we obtain local limit theorems for both finite and infinite mean, that is for the whole range $α\in (0,2)$. We also derive the asymptotics of the renewal function for $α\in (0,1]$.

math.PR↗
arXiv2020

On a conjecture of Seneta

In this short note we prove that $h_β(x) = β\int_0^x y^{β-1} \overline F(y) \mathrm{d} y$ is regularly varying with index $ρ\in [0,β)$ if and only if $V_β(x) = \int_{[0,x]} y^β\mathrm{d} F(y)$ is regularly varying with the same index. This implies an extended version of a recent conjecture by Seneta.

math.PR↗
arXiv2020

Limit laws for the norms of extremal samples

Let denote $S_n(p) = k_n^{-1} \sum_{i=1}^{k_n} \left( \log (X_{n+1-i,n} / X_{n-k_n, n}) \right)^p$, where $p > 0$, $k_n \leq n$ is a sequence of integers such that $k_n \to \infty$ and $k_n / n \to 0$, and $X_{1,n} \leq \ldots \leq X_{n,n}$ is the order statistics of iid random variables with regularly varying upper tail. The estimator $\widehat γ(n) = (S_n(p)/Γ(p+1))^{1/p}$ is an extension of the Hill estimator. We investigate the asymptotic properties of $S_n(p)$ and $\widehat γ(n)$ both for fixed $p > 0$ and for $p = p_n \to \infty$. We prove strong consistency and asymptotic normality under appropriate assumptions. Applied to real data we find that for larger $p$ the estimator is less sensitive to the change in $k_n$ than the Hill estimator.

math.ST↗
arXiv2020

Limit Theorems for Branching Processes with Immigration in a Random Environment

We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramér condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new results even in this very special setting.

math.PR↗
arXiv2019

Darling--Erdős theorem for Lévy processes at zero

We establish two equivalent versions of the Darling--Erdős theorem for Lévy processes in the domain of attraction of a stable process at zero with index $α\in(0,2)$. In the course of our proof we obtain a number of maximal and exponential inequalities for general Lévy processes, which should be of separate interest.

math.PR↗
arXiv2018

Convergence to stable limits for ratios of trimmed Levy processes and their jumps

We derive characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable process as t goes to 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in Kevei & Mason (2014) and produces new discrete distributions on the infinite simplex in the limit.

math.PR↗
arXiv2018

Darling-Kac theorem for renewal shifts in the absence of regular variation

We study null recurrent renewal Markov chains with renewal distribution in the domain of geometric partial attraction of a semistable law. Using the classical procedure of inversion, we derive a limit theorem similar to the Darling-Kac law along subsequences and obtain some interesting properties of the limit distribution. Also in this context, we obtain a Karamata type theorem along subsequences for positive operators. In both results, we identify the allowed class of subsequences. We provide several examples of nontrivial infinite measure preserving systems to which these results apply.

math.DS↗
arXiv2017

Regularly log-periodic functions and some applications

We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms.

math.PR↗
arXiv2016

Implicit renewal theory in the arithmetic case

We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution $X$ of the fixed point equations $X \stackrel{\mathcal{D}}{=} AX + B$, $X \stackrel{\mathcal{D}}{=} AX \vee B$ is $\ell (x) q(x) x^{-κ}$, where $q$ is a logarithmically periodic function $q(x e^h) = q(x)$, $x > 0$, with $h$ being the span of the arithmetic distribution of $\log A$, and $\ell$ is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevičius and Goldie.

math.PR↗
arXiv2016

A note on the Kesten--Grincevičius--Goldie theorem

Consider the perpetuity equation $X \stackrel{\mathcal{D}}{=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten--Grincevičius--Goldie theorem states that $P \{ X > x \} \sim c x^{-κ}$ if $E A^κ= 1$, $E A^κ\log_+ A < \infty$, and $E |B|^κ< \infty$. We assume that $E |B|^ν< \infty$ for some $ν> κ$, and consider two cases (i) $E A^κ= 1$, $E A^κ\log_+ A = \infty$; (ii) $E A^κ< 1$, $E A^t = \infty$ for all $t > κ$. We show that under appropriate additional assumptions on $A$ the asymptotic $P \{ X > x \} \sim c x^{-κ} \ell(x) $ holds, where $\ell$ is a nonconstant slowly varying function. We use Goldie's renewal theoretic approach.

math.PR↗
arXiv2016

Ergodic properties of generalized Ornstein--Uhlenbeck processes

We investigate ergodic properties of generalized Ornstein--Uhlenbeck processes. In particular, we provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster--Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.

math.PR↗
arXiv2015

High-frequency sampling of multivariate CARMA processes

High-frequency sampled multivariate continuous time autoregressive moving average processes are investigated. We obtain asymptotic expansion for the spectral density of the sampled MCARMA process $(Y_{nΔ})_{n \in \mathbb{Z}}$ as $Δ\downarrow 0$, where $(Y_t)_{t \in \mathbb{R}}$ is an MCARMA process. We show that the properly filtered process is a vector moving average process, and determine the asymptotic moving average representation of it, thus generalizing the results by Brockwell et al. in the univariate case to the multivariate model. The determination of the moving average representation of the filtered process, important for the analysis of high-frequency data, is difficult for any fixed positive $Δ$. However, the results established here provide a useful and insightful approximation when $Δ$ is very small.

math.PR↗
arXiv2015

On the Breiman conjecture

Let $Y_{1},Y_{2},\ldots $ be positive, nondegenerate, i.i.d. $G$ random variables, and independently let $X_{1},X_{2},\ldots $ be i.i.d. $F$ random variables. In this note we show that whenever $\sum X_{i}Y_{i}/\sum Y_{i}$ converges in distribution to nondegenerate limit for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, then $G$ necessarily belongs to the domain of attraction of a stable law with index less than 1. The class $\mathcal{F}$ contains those nondegenerate $X$ with a finite second moment and those $X$ in the domain of attraction of a stable law with index $1<α<2$.

math.PR↗