arXiv subjects

Martin Minchev

Publications and source records attributed to Martin Minchev.

9 recordsLinked to original sources

arXiv2026

Asymptotics for Beta-Splitting Trees via Homogeneous Fragmentations and Meromorphic Potential Theory

Inspired by recent work of Aldous, Janson, and Pittel on the critical beta-splitting model, we study the full beta-splitting family for beta greater than minus two through a canonical continuous-time embedding into a homogeneous exchangeable fragmentation. In this representation, the frequency of a tagged fragment is described by a subordinator. We express the continuous height of a typical leaf, its occupation probabilities, the discrete height, and the total continuous-time length in terms of the potential measure of this subordinator. Renewal theory yields first-order asymptotics and a central limit theorem for the continuous height. A regenerative-composition representation gives Gaussian limits for the discrete height above and at the critical value, and a non-Gaussian power-law limit below it. We also obtain residue expansions for the potential measure and the mean continuous height using meromorphic potential theory and generalized Nevanlinna functions. Finally, we study the maximum continuous-time height, proving a law of large numbers and a mixed Gumbel limit. At the critical parameter value, this resolves an open problem of Aldous and Janson.

math.PR↗

arXiv2026

On a moment determinacy conjecture of Bertoin and Yor

Let $\xi$ be an unkilled real-valued L\'evy process which drifts to $+\infty$ and has positive exponential moments of all orders, and define $I_\xi=\int_0^\infty e^{-\xi_t},dt$, and its reciprocal $X_\xi=1/I_\xi$. Bertoin and Yor proved that $X_\xi$ is moment-determinate when $\xi$ has no positive jumps, and conjectured that this condition is also necessary. We prove the latter. The proof is based on a lower bound near zero for the law of $I_\xi$. We show that a group of sufficiently many positive jumps near the origin puts $I_\xi$ on a suitable small scale. The first selected jump time is used as a one-dimensional smooth coordinate, yielding an absolutely continuous subcomponent of the law of $I_\xi$. After the change of variables, the resulting subdensity of $X_\xi$ satisfies a Krein moment indeterminacy criterion.

math.PR↗

arXiv2026

Preferential relocations enhance survival for Markov chains with killing

We investigate the impact on survival of a modification of the evolution of a sub-stochastic Markov chain that involves random relocations at previously visited states. Our central result is that such preferential relocations increase the persistence rate, meaning the survival probability decays more slowly than for the benchmark chain without relocations, and this improvement is strict under mild assumptions. We derive explicit lower bounds on the ratio of persistence rates when the relocation distribution is highly dispersed. The analysis relies on ergodic properties of so-called chains with complete connections, and a Feynman--Kac approach to estimate persistence.

math.PR↗

arXiv2025

Recent developments in exponential functionals of L\'evy processes

This survey aims to review two decades of progress on exponential functionals of (possibly killed) real-valued L\'evy processes. Since the publication of the seminal survey by Bertoin and Yor, substantial advances have been made in understanding the structure and properties of these random variables. At the same time, numerous applications of these quantities have emerged across various different contexts of modern applied probability. Motivated by all this, in this manuscript, we provide a detailed overview of these developments, beginning with a discussion of the class of special functions that have played a central role in recent progress, and then organising the main results on exponential functionals into thematic groups. Moreover, we complement several of these results and set them within a unified framework. Throughout, we strive to offer a coherent historical account of each contribution, highlighting both the probabilistic and analytical techniques that have driven the advances in the field.

math.PR↗

arXiv2025

On the probability of n equidistant points in high-dimensional lattices

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to \infty \quad \text{and} \quad m = \binom{n}{2}, \] with an explicit constant in terms of the first 4 moments of $X$. Moreover, we generalise this result to encompass all finitely supported $X$, as well as under different distances. Our method relies on the relatively rarely used multidimensional local limit theorem and an analysis of the lattice on $\mathbb{Z}^{\binom{n}{2}}$ spanned by the image of the \emph{overlapping} map \[ H : \{0,1\}^n \to \{0,1\}^{\binom{n}{2}}, \quad (v_1, \dots, v_n) \mapsto \Bigl( \mathbf{1}_{\{v_i \neq v_j\}} \Bigr)_{1 \le i < j \le n}. \]

math.PR↗

arXiv2024

Multi-type branching processes with immigration generated by point processes

Following the pivotal work of Sevastyanov, who considered branching processes with homogeneous Poisson immigration, much has been done to understand the behaviour of such processes under different types of branching and immigration mechanisms. Recently, the case where the times of immigration are generated by a non-homogeneous Poisson process was considered in depth. In this work, we try to demonstrate how one can use the framework of point processes in order to go beyond the Poisson process. As an illustration, we show how to transfer techniques from the case of Poisson immigration to the case where it is spanned by a determinantal point process.

math.PR↗

arXiv2024

BFS versus DFS for fixed-level targets in ordered trees

We find the average time complexity of the breadth-first search (BFS) and the depth-first search (DFS) algorithms, when one searches for a target node selected uniformly at random among all nodes at level $\ell$ in the set of ordered trees with $n$ edges. Intuition suggests that on average BFS must be asymptotically faster than DFS if and only if $\ell$, as a function of $n$, is below a certain threshold. We confirm this intuition by showing that there exists a unique constant $\lambda\approx 0.789004$, such that in expectation BFS is asymptotically faster than DFS if and only if $\ell\leq \lambda\sqrt{n}$. This gives us a practical rule to select between the two algorithms, even when we do not know the exact value of $\ell$, but only an estimate of it. Furthermore, we find the asymptotic average time complexity of BFS in the given setting for an arbitrary class of Galton--Watson trees, which includes ordered trees, binary trees, and other popular classes. We use results on the occupation measure of Brownian excursions, as well as combinatorial identities related to lattice paths. Finally, we consider the simple \textit{truncated DFS} algorithm, which can be shown easily to be asymptotically faster than both BFS and DFS when $\ell$ is known in advance. We show that in fact its asymptotic time complexity is $1/2$ of the asymptotic complexity of BFS, when $\ell = s\sqrt{n}$ for any constant $s$. Several further questions are also raised.

cs.DS↗

arXiv2023

Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of L\'evy processes

Let $\xi$ be a L\'{e}vy process and $I_\xi(t):=\int_{0}^te^{-\xi_s}\mathrm{d} s$, $t\geq 0,$ be the exponential functional of L\'{e}vy processes on deterministic horizon. Given that $\lim_{t\to \infty}\xi_t=-\infty$ we evaluate for general functions $F$ an upper bound on the rate of decay of $\mathbb{E}\left(F(I_\xi(t))\right)$ based on an explicit integral criterion. When $\mathbb{E}\left(\xi_1\right)\in\left(-\infty,0\right)$ and $\mathbb{P}\left(\xi_1>t\right)$ is regularly varying of index $\alpha>1$ at infinity, we show that the law of $I_\xi(t)$, suitably normed and rescaled, converges weakly to a probability measure stemming from a new generalisation of the product factorisation of classical exponential functionals. These results substantially improve upon the existing literature and are obtained via a novel combination between Mellin inversion of the Laplace transform of $\mathbb{E}\left(I^{-a}_{\xi}(t)\mathbf{1}_{\left\{I_{\xi}(t)\leq x\right\}}\right)$, $a\in (0,1)$, $x\in(0,\infty],$ and Tauberian theory augmented for integer-valued $\alpha$ by a suitable application of the one-large jump principle in the context of the de Haan theory. The methodology rests upon the representation of the aforementioned Mellin transform in terms of the recently introduced bivariate Bernstein-gamma functions for which we develop the following new results of independent interest (for general $\xi$): we link these functions to the $q$-potentials of $\xi$; we show that their derivatives at zero are finite upon the finiteness of the aforementioned integral criterion; we offer neat estimates of those derivatives along complex lines. These results are useful in various applications of the exponential functionals themselves and in different contexts where properties of bivariate Bernstein-gamma functions are needed. $\xi$ need not be non-lattice.

math.PR↗

arXiv2023

Asymptotics for densities of exponential functionals of subordinators

In this paper we derive non-classical Tauberian asymptotic at infinity for the tail, the density and the derivatives thereof of a large class of exponential functionals of subordinators. More precisely, we consider the case when the Lévy measure of the subordinator satisfies the well-known and mild condition of positive increase. This is achieved via a convoluted application of the saddle point method to the Mellin transform of these exponential functionals which is given in terms of Bernstein-gamma functions. To apply the saddle point method we improved the Stirling type of asymptotic for Bernstein-gamma functions and the latter is of interest beyond this paper as the Bernstein-gamma functions are applicable in different settings especially through their asymptotic behaviour in the complex plane. As an application we have derived the asymptotic of the density and its derivatives for all exponential functionals of non-decreasing, potentially compound Poisson processes which turns out to be precisely as that of an exponentially distributed random variable. We show further that a large class of densities are even analytic in a cone of the complex plane.

math.PR↗