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Matyas Barczy

Publications and source records attributed to Matyas Barczy.

At least 19 recordsLinked to original sources
arXiv2026

Expected loss of quasi-arithmetic means of exchangeable random variables

The purpose of this paper is to establish certain relationships between the theories of means (aggregation functions) and loss functions. Namely, we derive sufficient conditions under which the sequence of the expected losses of a quasi-arithmetic mean of the first $n$ members from a sequence of exchangeable random variables is (strictly) decreasing with respect to $n$. Some new properties of functions which are strongly convex with respect to a nonnegative even error function, such as a Jensen-type inequality, are described. Additionally, non-trivial new examples for such functions are presented as well.

math.PR↗
arXiv2025

Distributional properties of first jump times of CBI processes with jump sizes in given Borel sets

We derive an expression for the joint distribution function of the first jump times of a continuous state and continuous time branching process with immigration (CBI process) with jump sizes in given Borel sets having finite total L\'evy measures, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the CBI process in question. Our result generalizes a corresponding result of He and Li (2016), who considered this problem in case of a single Borel set having finite total L\'evy measure.

math.PR↗
arXiv2025

Jensen convex functions and doubly stochastic matrices

Given an nxn doubly stochastic matrix P satisfying an appropriate condition of linear algebraic-type, and a function f defined on a nonempty interval, we show that the validity of a convexity-type functional inequality for f in terms P implies that f is Jensen convex. We also prove that if f is convex, then the functional inequality in question holds for all doubly stochastic matrices of any order. The particular case when the doubly stochastic matrix is a circulant one is also considered.

math.CA↗
arXiv2025

A convexity-type functional inequality with infinite convex combinations

Given a function $f$ defined on a nonempty and convex subset of the $d$-dimensional Euclidean space, we prove that if $f$ is bounded from below and it satisfies a convexity-type functional inequality with infinite convex combinations, then $f$ has to be convex. We also give alternative proofs of a generalization of some known results on convexity with infinite convex combinations due to Dar\'oczy and P\'ales (1987) and Pavi\'c (2019) using a probabilistic version of Jensen inequality.

math.CA↗
arXiv2025

A series of definite integrals involving upper incomplete Gamma functions

Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial role. We also give an interesting application of our result, namely, a new summation formula for some derivatives of the Bessel functions of the first kind and the Struve functions with respect to the order.

math.CA↗
arXiv2024

Monotone representation and measurability of generalized $\psi$-estimators

We investigate the monotone representation and measurability of generalized $\psi$-estimators introduced by the authors in 2022. Our first main result, applying the unique existence of a generalized $\psi$-estimator, allows us to construct this estimator in terms of a function $\psi$, which is decreasing in its second variable. We then interpret this result as a bridge from a nonconvex optimization problem to a convex one. Further, supposing that the underlying measurable space (sample space) has a measurable diagonal and some additional assumptions on $\psi$, we show that the measurability of a generalized $\psi$-estimator is equivalent to the measurability of the corresponding function $\psi$ in its first variable.

math.ST↗
arXiv2024

A stochastic approach in physics exercises of mathematics education

We present a method for incorporating a stochastic point of view into physics exercises of mathematics education. The core of our method is the randomization of some inputs, the system model used does not differ from what we would use in the deterministic approach. We consider exercises from the theory of projectile motion and statics. The outputs of stochastic models are random variables, and we usually determine their probability distributions, expected values, variances, and relative standard deviations, and the probabilities of some events related to them are also calculated. Students and teachers familiar with elementary probability theory and mechanics may find these exercises useful for understanding some basic concepts of stochastic mechanics.

physics.ed-ph↗
arXiv2024

Axiomatic characterisation of generalized $\psi$-estimators

We give axiomatic characterisations of generalized $\psi$-estimators and (usual) $\psi$-estimators (also called $Z$-estimators), respectively. The key properties of estimators that come into play in the characterisation theorems are the symmetry, the (strong) internality and the asymptotic idempotency. In the proofs, a separation theorem for Abelian subsemigroups plays a crucial role.

math.ST↗
arXiv2024

On H\"older continuity and $p^\mathrm{th}$-variation function of Weierstrass-type functions

We study H\"older continuity, $p^\mathrm{th}$-variation function and Riesz variation of Weierstrass-type functions along the sequence of $b$-adic partitions, where $b>1$ is an integer. By a Weierstrass-type function, we mean that in the definition of the well-known Weierstrass function, the power function is replaced by a submultiplicative function, and the Lipschitz continuous cosine and sine functions are replaced by a general periodic H\"older continuous function.

math.CA↗
arXiv2024

Asymptotic behavior of some strongly critical decomposable 3-type Galton--Watson processes with immigration

We study the asymptotic behavior of a critical decomposable 3-type Galton-Watson process with immigration when its offspring mean matrix is triangular with diagonal entries 1. 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 such a Galton-Watson process converges weakly. The limit process can be described using independent squared Bessel processes $({\mathcal X}_{t,1})_{t\geq0}$, $({\mathcal X}_{t,2})_{t\geq0}$, and $({\mathcal X}_{t,3})_{t\geq0}$, the linear combinations of the integral processes of $({\mathcal X}_{t,1})_{t\geq0}$ and $({\mathcal X}_{t,2})_{t\geq0}$, and possibly the 2-fold iterated integral process of $({\mathcal X}_{t,1})_{t\geq0}$. The presence of the 2-fold iterated integral process in the limit distribution is a new phenomenon in the description of asymptotic behavior of critical multi-type Galton-Watson processes with immigration. Our results complete and extend some results of Foster and Ney (1978) for some strongly critical decomposable 3-type Galton-Watson processes with immigration.

math.PR↗
arXiv2024

Determining classes for generalized $\psi$-estimators

We prove that the values of a generalized $\psi$-estimator (introduced by Barczy and P\'ales in 2025) on samples of arbitrary length but having only two different observations uniquely determine the values of the estimator on any sample of arbitrary length without any restriction on the number of different observations. In other words, samples of arbitrary length but having only two different observations form a determining class for generalized $\psi$-estimators. We also obtain a similar statement for the comparison of generalized $\psi$-estimators using comparative functions, and, as a corollary of this result, we derive the Schweitzer's inequality (also called Kantorovich's inequality).

math.ST↗
arXiv2024

Basic properties of generalized $\psi$-estimators

We establish several properties of (weighted) generalized $\psi$-estimators introduced by Barczy and P\'ales in 2022: mean-type, monotonicity and sensitivity properties, bisymmetry-type inequality and some asymptotic and continuity properties as well. We also illustrate these properties by providing several examples including statistical ones as well.

math.ST↗
arXiv2023

Comparison and equality of generalized $\psi$-estimators

We solve the comparison problem for generalized $\psi$-estimators introduced in Barczy and P\'ales (2022). Namely, we derive several necessary and sufficient conditions under which a generalized $\psi$-estimator less than or equal to another $\psi$-estimator for any sample. We also solve the corresponding equality problem for generalized $\psi$-estimators. For applications, we solve the two problems in question for Bajraktarevi\'c-type- and quasi-arithmetic-type estimators. We also apply our results for some known statistical estimators such as for empirical expectiles and Mathieu-type estimators and for solutions of likelihood equations in case of normal, a Beta-type, Gamma, Lomax (Pareto type II), lognormal and Laplace distributions.

math.ST↗
arXiv2023

Distributional properties of jumps of multi-type CBI processes

We study the distributional properties of jumps of multi-type continuous state and continuous time branching processes with immigration (multi-type CBI processes). We derive an expression for the distribution function of the first jump time of a multi-type CBI process with jump size in a given Borel set having finite total L\'evy measure, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the multi-type CBI process in question. Using this we derive an expression for the distribution function of the local supremum of the norm of the jumps of a multi-type CBI process. Further, we show that if $A$ is a nondegenerate rectangle anchored at zero and with total L\'evy measure zero, then the probability that the local coordinate-wise supremum of jumps of the multi-type CBI process belongs to $A$ is zero. We also prove that a converse statement holds.

math.PR↗
arXiv2023

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↗
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↗