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Matyas Barczy

Publications and source records attributed to Matyas Barczy.

At least 37 records · Page 2Linked to original sources
arXiv2024

Diffusion approximation of critical controlled multi-type branching processes

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.

math.PR↗
arXiv2023

Mixing convergence of LSE for supercritical AR(2) processes with Gaussian innovations using random scaling

We prove mixing convergence of the least squares estimator of autoregressive parameters for supercritical autoregressive processes of order 2 with Gaussian innovations having real characteristic roots with different absolute values. We use an appropriate random scaling such that the limit distribution is a two-dimensional normal distribution concentrated on a one-dimensional ray determined by the characteristic root having the larger absolute value.

math.ST↗
arXiv2022

A multidimensional stable limit theorem

We establish multidimensional analogues of one-dimensional stable limit theorems due to Häusler and Luschgy (2015) for so called explosive processes. As special cases we present multidimensional stable limit theorems involving multidimensional normal-, Cauchy- and stable distributions as well.

math.PR↗
arXiv2022

Limit theorems for deviation means of independent and identically distributed random variables

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables (for the strong law of large numbers, we suppose only pairwise independence instead of (total) independence). The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well-studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. Especially, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (2016) and the ones for Bajraktarević means due to Barczy and Burai (2022).

math.PR↗
arXiv2022

Probability equivalent level of Value at Risk and higher-order Expected Shortfalls

We investigate the probability equivalent level of Value at Risk and $n^{\mathrm{th}}$-order Expected Shortfall (called PELVE_n), which can be considered as a variant of the notion of the probability equivalent level of Value at Risk and Expected Shortfall (called PELVE) due to Li and Wang (2022). We study the finiteness, uniqueness and several properties of PELVE_n, we calculate PELVE_n of some notable distributions, PELVE_2 of a random variable having generalized Pareto excess distribution, and we describe the asymptotic behaviour of PELVE_2 of regularly varying distributions as the level tends to $0$. Some properties of $n^{\mathrm{th}}$-order Expected Shortfall are also investigated. Among others, it turns out that the Gini Shortfall at some level $p\in[0,1)$ corresponding to a (loading) parameter $λ\geq 0$ is the linear combination of the Expected Shortfall at level $p$ and the $2^{\mathrm{nd}}$-order Expected Shortfall at level $p$ with coefficients $1-2λ$ and $2λ$, respectively.

q-fin.RM↗
arXiv2022

Asymptotic behaviour of critical decomposable 2-type Galton-Watson processes with immigration

In this paper the asymptotic behaviour of a critical 2-type Galton-Watson process with immigration is described when its offspring mean matrix is reducible, in other words, when the process is decomposable. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from a critical decomposable 2-type Galton-Watson process with immigration converges weakly. The limit process can be described using one or two independent squared Bessel processes and possibly the unique stationary distribution of an appropriate single-type subcritical Galton-Watson process with immigration. Our results complete and extend the results of Foster and Ney (1978) for some strongly critical decomposable 2-type Galton-Watson processes with immigration.

math.PR↗
arXiv2021

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

We provide a generalization of Theorem 1 in Bartkiewicz, Jakubowski, Mikosch and Wintenberger (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton--Watson branching process with regularly varying immigration having index in $(0, 1) \cup (1, 4/3)$ in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

math.PR↗
arXiv2021

Random means generated by random variables: expectation and limit theorems

We introduce the notion of a random mean generated by a random variable and give a construction of its expected value. We derive some sufficient conditions under which strong laws of large numbers and some limit theorems hold for random means generated by the elements of a sequence of independent and identically distributed random variables.

math.PR↗
arXiv2021

A note on asymptotic behavior of critical Galton-Watson processes with immigration

In this somewhat didactic note we give a detailed alternative proof of the known result due to Wei and Winnicki (1989) which states that under second order moment assumptions on the offspring and immigration distributions the sequence of appropriately scaled random step functions formed from a critical Galton-Watson process with immigration (starting from not necessarily zero) converges weakly towards a squared Bessel process. The proof of Wei and Winnicki (1989) is based on infinitesimal generators, while we use limit theorems for random step processes towards a diffusion process due to Ispány and Pap (2010). This technique was already used in Ispány (2008), where he proved functional limit theorems for a sequence of some appropriately normalized nearly critical Galton-Watson processes with immigration starting from zero, where the offspring means tend to its critical value 1. As a special case of Theorem 2.1 in Ispány (2008) one can get back the result of Wei and Winnicki (1989) in the case of zero initial value. In the present note we handle non-zero initial values with the technique used in Ispány (2008), and further, we simplify some of the arguments in the proof of Theorem 2.1 in Ispány (2008) as well.

math.PR↗
arXiv2021

Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables

We derive strong laws of large numbers and central limit theorems for Bajraktarević, Gini and exponential- (also called Beta-type) and logarithmic Cauchy quotient means of independent identically distributed (i.i.d.) random variables. The exponential- and logarithmic Cauchy quotient means of a sequence of i.i.d. random variables behave asymptotically normal with the usual square root scaling just like the geometric means of the given random variables. Somewhat surprisingly, the multiplicative Cauchy quotient means of i.i.d. random variables behave asymptotically in a rather different way: in order to get a non-trivial normal limit distribution a time dependent centering is needed.

math.PR↗
arXiv2021

Asymptotic behavior of projections of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a fourth order moment condition on the branching and a second order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous state and continuous time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In case of a non-trivial process, under a first order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

math.PR↗
arXiv2020

On approximations of Value at Risk and Expected Shortfall involving kurtosis

We derive new approximations for the Value at Risk and the Expected Shortfall at high levels of loss distributions with positive skewness and excess kurtosis, and we describe their precisions for notable ones such as for exponential, Pareto type I, lognormal and compound (Poisson) distributions. Our approximations are motivated by that kind of extensions of the so-called Normal Power Approximation, used for approximating the cumulative distribution function of a random variable, which incorporate not only the skewness but the kurtosis of the random variable in question as well. We show the performance of our approximations in numerical examples and we also give comparisons with some known ones in the literature.

q-fin.RM↗
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↗
arXiv2020

On simultaneous limits for aggregation of stationary randomized INAR(1) processes with Poisson innovations

We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $α\in(0,1)$ and with idiosyncratic Poisson innovations. Assuming that $α$ has a density function of the form $ψ(x) (1 - x)^β$, $x \in (0,1)$, with $β\in(-1,\infty)$ and $\lim_{x\uparrow 1} ψ(x) = ψ_1 \in (0,\infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $β\in(-1,0]$ in the so-called simultaneous case, i.e., when both $N$ and the time scale $n$ increase to infinity at a given rate. The case $β\in(0,\infty)$ remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

math.PR↗
arXiv2020

On tail behaviour of stationary second-order Galton-Watson processes with immigration

A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigration) at any fixed time is regularly varying provided that the initial sizes of the population are independent and regularly varying.

math.PR↗
arXiv2020

A new example for a proper scoring rule

We give a new example for a proper scoring rule motivated by the form of Anderson--Darling distance of distribution functions and Example 5 in Brehmer and Gneiting (2020).

math.ST↗
arXiv2020

On aggregation of subcritical Galton-Watson branching processes with regularly varying immigration

We study an iterated temporal and contemporaneous aggregation of $N$ independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index $α\in (0, 2)$. Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to exist when first taking the limit as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is an $α$-stable process if $α\in (0, 1) \cup (1, 2)$, and a deterministic line with slope $1$ if $α= 1$.

math.PR↗