Searcharxiv⌕ Search

arXiv subjects

Konrad Kolesko

Publications and source records attributed to Konrad Kolesko.

At least 19 recordsLinked to original sources
arXiv2026

Maximal and minimal displacement of supercritical branching random walks on free products of groups

We prove that the maximal and minimal displacement of branching random walks with mean offspring number $ρ>1$ on free products of finite groups grows linearly almost surely. More precisely, we establish that the linear speed for the maximal (respectively minimal) displacement is given by the largest (respectively smallest) intersection point of the large deviation rate function of the underlying random walk with the horizontal line at height $\logρ$. The proof is based on constructing an associated multitype branching process which consists of particles that travel fast enough, and distinguishing the types via the suffix of the particles locations.

math.PR↗
arXiv2026

Asymptotic fluctuations in supercritical multi-type Crump-Mode-Jagers processes

We consider an irreducible, supercritical multi-type general branching process (Crump-Mode-Jagers process) with finite type space, counted with random characteristic. We prove that, under certain second moment assumptions, a central limit theorem holds for the process. Our result extends the single-type central limit theorem obtained recently in [Ann. Probab. 52 (2024), no. 4, 1538-1606] and unifies various limiting results for specific branching processes.

math.PR↗
arXiv2025

Explosion of Crump-Mode-Jagers processes with critical immediate offspring

We study the phenomenon of explosion in general (Crump-Mode-Jagers) branching processes, which refers to the event where an infinite number of individuals are born in finite time. In a critical setting where the expected number of immediate offspring per individual is exactly one, whether or not explosion occurs depends on the fine properties of the reproduction point process. We provide two sufficient conditions for explosion in these CMJ processes. The first uses a comparison with Galton-Watson processes in varying environments, while the second relies on a comparison with Bellman-Harris branching processes. Our main result is an equivalent characterization of explosion, expressed as an integral test, in the case where the reproduction point process is Poisson. For the derivation, we also study the fixed-point equation associated with a smoothing transform, which is known to describe the distribution of the explosion time. We use multiplicative martingales to show that this distribution is an attractive fixed point of the smoothing transform, which in particular implies its uniqueness modulo an additive shift.

math.PR↗
arXiv2025

Convergence of complex martingales in supercritical multi-type general branching processes in $L^q$ for $1 < q \leq 2$

Nerman's martingale plays a central role in the law of large numbers for both, single- and multi-type, supercritical general branching processes. There are further, complex-valued Nerman-type martingales in the single-type process that figure in the finer fluctuations of these processes. We construct the analogous martingales for the process with finitely many types and give sufficient conditions for these martingales to converge in $L^q$ for $q \in (1,2]$.

math.PR↗
arXiv2025

Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes

We consider the Markov renewal equation $F(t) = f(t) + \boldsymbolμ*F(t)$ for vector-valued functions $f,F: \mathbb{R} \to \mathbb{R}^{p}$ and a $p \times p$ matrix $\boldsymbolμ$ of locally finite measures $μ^{i,j}$ on $[0,\infty)$, $i,j=1,\ldots,p$. Sgibnev [Semimultiplicative estimates for the solution of the multidimensional renewal equation. {\em Izv.\ Ross.\ Akad.\ Nauk Ser.\ Mat.}, 66(3):159--174, 2002] derived an asymptotic expansion for the solution $F$ to the above equation. We give a new, more elementary proof of Sgibnev's result, which also covers the reducible case. As a corollary, we infer an asymptotic expansion for the mean of a multi-type general branching process with finite type space counted with random characteristic. Finally, some examples are discussed that illustrate phenomena of multi-type branching.

math.PR↗
arXiv2024

Gaussian fluctuations for the two urn model

We introduce a modification of the generalized Pólya urn model containing two urns and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$, $J\in\mathbb{N}$ added to the urns after $n$ draws. We provide sufficient conditions under which the random variables $(B_j(n))_{n\in\mathbb{N}}$ properly normalized and centered converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second order behavior occurs at $\sqrtρ$ and not at $ρ/2$, where $ρ$ is the dominant eigenvalue of the mean replacement matrix.

math.PR↗
arXiv2024

Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^φ)_{t \geq 0}$ counted with a random characteristic $φ$. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, $e^{-αt} \mathcal Z_t^φ$ converges almost surely as $t \to \infty$ to $aW$. Here, $α>0$ is the Malthusian parameter, $a$ is a constant and $W$ is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for $(\mathcal Z_t^φ)_{t \geq 0}$. More precisely, we show that there exist a constant $k \in \mathbb N_0$ and a function $H(t)$, a finite random linear combination of functions of the form $t^j e^{λt}$ with $α/2 \leq \mathrm{Re}(λ)<α$, such that $(\mathcal Z_t^φ- a e^{αt}W -H(t))/\sqrt{t^k e^{αt}}$ converges in distribution to a normal random variable with random variance. This result unifies and extends various central limit theorem-type results for specific branching processes.

math.PR↗
arXiv2023

Limit theorems for discrete multitype branching processes counted with a characteristic

For a discrete time multitype supercritical Galton-Watson process $(Z_n)_{n\in \mathbb{N}}$ and corresponding genealogical tree $\mathbb{T}$, we associate a new discrete time process $(Z_n^Φ)_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^Φ$ is determined by a (random) characteristic $Φ$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^Φ$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $Φ_u$, where $(Φ_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $Φ$. Such processes are known in the literature under the name of Crump-Mode-Jagers (CMJ) processes counted with characteristic $Φ$. We derive a LLN and a CLT for the process $(Z_n^Φ)_{n\in\mathbb{N}}$ in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten-Stigum [17].

math.PR↗
arXiv2021

Gaussian fluctuations and a law of the iterated logarithm for Nerman's martingale in the supercritical general branching process

In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale $(W_t)_{t \geq 0}$ to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (a.k.a. Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value $W$ figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman's martingale, converges to its limit $W$. More precisely, assuming the existence of a Malthusian parameter $α> 0$ and $W_0\in L^2$, we prove a functional central limit theorem for $(W-W_{t+s})_{s\in\mathbb{R}}$, properly normalized, as $t\to\infty$. The weak limit is a randomly scaled time-changed Brownian motion. Under an additional technical assumption, we prove a law of the iterated logarithm for $W-W_t$.

math.PR↗
arXiv2020

Solutions to kinetic-type evolution equations: beyond the boundary case

We study the asymptotic behavior as $t \to \infty$ of a time-dependent family $(μ_t)_{t \geq 0}$ of probability measures on $\mathbb{R}$ solving the kinetic-type evolution equation $\partial_t μ_t + μ_t = Q(μ_t)$ where $Q$ is a smoothing transformation on $\mathbb{R}$. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928-1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677-693, 2020]. Combining the refined analysis of the latter paper, which provides a probabilistic description of the solution $μ_t$ as the law of a suitable random sum related to a continuous-time branching random walk at time $t$, with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the remaining case that has been left open until now. In the course of our work, we significantly weaken the assumptions in the literature that guarantee the existence (and uniqueness) of a solution to the evolution equation $\partial_t μ_t + μ_t = Q(μ_t)$.

math.PR↗
arXiv2018

Fluctuations of Biggins' martingales at complex parameters

The long-term behavior of a supercritical branching random walk can be described and analyzed with the help of Biggins' martingales, parametrized by real or complex numbers. The study of these martingales with complex parameters is a rather recent topic. Assuming that certain sufficient conditions for the convergence of the martingales to non-degenerate limits hold, we investigate the fluctuations of the martingales around their limits. We discover three different regimes. First, we show that for parameters with small absolute values, the fluctuations are Gaussian and the limit laws are scale mixtures of the real or complex standard normal laws. We also cover the boundary of this phase. Second, we find a region in the parameter space in which the martingale fluctuations are determined by the extremal positions in the branching random walk. Finally, there is a critical region (typically on the boundary of the set of parameters for which the martingales converge to a non-degenerate limit) where the fluctuations are stable-like and the limit laws are the laws of randomly stopped Lévy processes satisfying invariance properties similar to stability.

math.PR↗
arXiv2018

Absolute continuity of the martingale limit in branching processes in random environment

We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $ξ=(ξ_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|ξ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $ξ$ the law of $W$ conditioned on the environment $ξ$ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.

math.PR↗
arXiv2018

Local fluctuations of critical Mandelbrot cascades

We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a~sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $μ$ fluctuates. For any given point $x$, denoting by $B_n(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $μ(B_n(x))$ as $n \to \infty$.

math.PR↗
arXiv2017

Stable-like fluctuations of Biggins' martingales

Let $(W_n(θ))_{n \in \mathbb{N}_0}$ be Biggins' martingale associated with a supercritical branching random walk, and let $W(θ)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1(θ)$ belongs to the domain of normal attraction of an $α$-stable distribution for some $α\in (1,2)$, then, as $n\to\infty$, there is weak convergence of the tail process $(W(θ) - W_{n-k}(θ))_{k \in \mathbb{N}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $α$-stable marginals.

math.PR↗
arXiv2016

Convergence of complex martingales in the branching random walk: the boundary

Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $λ$ from an open set $Λ\subseteq \mathbb{C}^d$. We investigate the martingales corresponding to parameters from the boundary $\partial Λ$ of $Λ$. The boundary can be decomposed into several parts. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the remaining part, there is convergence to a non-degenerate limit. The arguments that give this convergence also apply in $Λ$ and require weaker moment assumptions than the ones used by Biggins.

math.PR↗
arXiv2014

Fixed Points of the Multivariate Smoothing Transform: The Critical Case

Given a sequence $(T_1, T_2, ...)$ of random $d \times d$ matrices with nonnegative entries, suppose there is a random vector $X$ with nonnegative entries, such that $ \sum_{i \ge 1} T_i X_i $ has the same law as $X$, where $(X_1, X_2, ...)$ are i.i.d. copies of $X$, independent of $(T_1, T_2, ...)$. Then (the law of) $X$ is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case $d=1$, a function $m$ is introduced, such that the existence of $α\in (0,1]$ with $m(α)=1$ and $m'(α) \le 0$ guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case $m'(α)=0$ and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit.

math.PR↗
arXiv2014

Brownian Motion on graph-like spaces

We construct Brownian motion on a wide class of metric spaces similar to graphs, and show that its cover time admits an upper bound depending only on the length of the space.

math.PR↗