arXiv subjects

Matyas Barczy

Publications and source records attributed to Matyas Barczy.

At least 19 recordsLinked to original sources

arXiv2026

Pick-up sticks problems for exponential and Pareto distributions

We deal with the following two pick-up sticks problems: given an integer n greater than or equal to 3, we choose n sticks with independent random lengths from an exponential distribution or Pareto type I distribution, and we are interested in the probability that no three of them can be the sides of a triangle. In case of exponential distribution, this problem has been solved by other authors: the probability in question is equal to $n!/\big((F_3-1)(F_4-1)\cdots(F_{n+2}-1)\big)$, where $(F_i)_{i=1}^\infty$ is the sequence of Fibonacci numbers. We give a new, self-contained and detailed proof for this formula, which uses a different method compared to the earlier ones. In case of Pareto type I distribution, we derive new formulae for the corresponding probability.

math.PR↗

arXiv2026

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

We study the asymptotic behavior of a critical decomposable multi-type Galton-Watson process with immigration whose offspring mean matrix is triangular with diagonal entries 1. Under second-order moment conditions on the offspring and immigration distributions, we establish a functional limit theorem for the sequence of appropriately scaled processes, and describe the limit process as independent squared Bessel processes and some linear combinations of their iterated integral processes. Our work completes and extends some results of Foster and Ney (1978) for some strongly critical decomposable multi-type Galton-Watson processes with immigration.

math.PR↗

arXiv2026

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 Lévy measures, where by total Lévy measure, we mean 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 Lévy measure.

math.PR↗

arXiv2026

Functional limit theorems for Galton-Watson processes with inhomogeneous immigration towards a deterministic limit function

We study the asymptotic behavior of a sequence of Galton-Watson processes with inhomogeneous immigration when the limit of the means of the offspring distributions is equal to 1. Under a growth condition on the variances of the offspring distributions, assuming the locally uniform convergence of the expectation functions of properly scaled immigration processes towards a càdlàg function, and the locally uniform convergence of the mean absolute deviation functions of the scaled immigration processes in question towards zero, we establish a functional limit theorem for the sequence of Galton-Watson processes with inhomogeneous immigration in question towards a deterministic limit process. We also provide some applications, among others, we show that a recent result of Rahimov and Sharipov (2024) on a functional limit theorem for a sequence of nearly critical Galton-Watson processes with dependent immigration can be recovered using our result.

math.PR↗

arXiv2026

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

We study Hölder 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 Hölder continuous function.

math.CA↗

arXiv2026

Functional limit theorems for Galton--Watson processes with inhomogeneous immigration

We study the asymptotic behavior of a sequence of Galton--Watson processes with inhomogeneous immigration when the limit of the means of the offspring distributions is less than $1$ or equal to $1$. Under growth conditions on the expected values of the immigration distributions and the variances of the offspring distributions, and assuming the weak convergence of properly scaled immigration processes towards a non-negative stochastic process $\mathcal Y$ with càdlàg or continuous sample paths, we establish functional limit theorems for the sequence of Galton--Watson processes with inhomogeneous immigration in question. The limit stochastic processes can be represented as a constant multiple or an integral functional of $\mathcal Y$ .

math.PR↗

arXiv2026

On variation functions of lacunary series

We study variation functions of some lacunary series along the sequence of $b$-adic partitions, where $b>1$ is an integer. By such a lacunary series, we mean that in the definition of the well-known Weierstrass function, the Lipschitz continuous cosine and sine functions are replaced by a continuous and piecewise continuously differentiable periodic function whose Fourier series has zero $(jb)^{\mathrm{th}}$-Fourier coefficients for all non-zero integers $j$. In the proofs, we use a probabilistic approach applying a martingale central limit theorem. Our result extends some recent results of Han, Schied and Zhang (2021), and can be applied when one chooses the periodic function in question, for example, as the Takagi function, a Weierstrass function or a certain lacunary series itself.

math.PR↗

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↗

arXiv2026

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↗

arXiv2026

Monotone representation and measurability of generalized $ψ$-estimators

We investigate the monotone representation and measurability of generalized $ψ$-estimators introduced by the authors in 2022. Our first main result, applying the unique existence of a generalized $ψ$-estimator, allows us to construct this estimator in terms of a function $ψ$, 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 $ψ$, we show that the measurability of a generalized $ψ$-estimator is equivalent to the measurability of the corresponding function $ψ$ in its first variable.

math.ST↗

arXiv2026

Axiomatic characterisation of generalized $ψ$-estimators

We give axiomatic characterisations of generalized $ψ$-estimators and (usual) $ψ$-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↗

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 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↗

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 Daróczy and Páles (1987) and Pavić (2019) using a probabilistic version of Jensen inequality.

math.CA↗

arXiv2025

Determining classes for generalized $ψ$-estimators

We prove that the values of a generalized $ψ$-estimator (introduced by Barczy and Páles 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 $ψ$-estimators. We also obtain a similar statement for the comparison of generalized $ψ$-estimators using comparative functions, and, as a corollary of this result, we derive the Schweitzer's inequality (also called Kantorovich's inequality).

math.ST↗

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↗

arXiv2025

Existence and uniqueness of weighted generalized $ψ$-estimators

We introduce the notions of generalized and weighted generalized $ψ$-estimators as unique points of sign change of some appropriate functions, and we give necessary as well as sufficient conditions for their existence. We also derive a set of sufficient conditions under which the so-called $ψ$-expectation function has a unique point of sign change. We present several examples from statistical estimation theory, where our results are well-applicable. For example, we consider the cases of empirical quantiles, empirical expectiles, some $ψ$-estimators that are important in robust statistics, and some examples from maximum likelihood theory as well. Further, we introduce Bajraktarević-type (in particular, quasi-arithmetic-type) $ψ$-estimators. Our results specialized to $ψ$-estimators with a function $ψ$ being continuous in its second variable provide new results for (usual) $ψ$-estimators (also called Z-estimators).

math.ST↗

arXiv2024

Comparison and equality of generalized $ψ$-estimators

We solve the comparison problem for generalized $ψ$-estimators introduced in Barczy and Páles (2022). Namely, we derive several necessary and sufficient conditions under which a generalized $ψ$-estimator less than or equal to another $ψ$-estimator for any sample. We also solve the corresponding equality problem for generalized $ψ$-estimators. For applications, we solve the two problems in question for Bajraktarević-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↗