arXiv subjects

Anna Talarczyk

Publications and source records attributed to Anna Talarczyk.

At least 19 recordsLinked to original sources

arXiv2026

Large time behavior of critical marked Hawkes processes with heavy tailed marks and related branching particle systems

We study large time behavior of critical marked Hawkes processes and related branching particle systems. In case of marked Hawkes processes we assume that the kernel function has multiplicative form and the marks corresponding to the events are nonnegative and are assigned independently from a common distribution. This distribution is in the normal domain of attraction of a $(1+\beta)$-stable law with $0<\beta<1$. Moreover, we assume that the mean number of events triggered by a single event is equal to $1$ (criticality). We show that, as the time is speeded up, if $\beta$ is small enough then, the event counting process, appropriately normalized, converges to a spectrally positive $1/(1+\beta)$ stable L\'evy process. The convergence holds in law in the Skorokhod space of c\`adl\`ag functions equipped with $M_1$ topology. We also study a borderline case. The present paper complements the results of [A.Talarczyk:``A generalized central limit theorem for critical marked Hawkes processes'', arXiv:2504.11612], where the same model was studied in case of ``large'' $\beta$. We employ techniques involving a branching representation of marked Hawkes processes. This approach allows to study more general branching processes with branching mechanism in the normal domain of attraction of $(1+\beta)$-stable law.

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arXiv2025

A generalized central limit theorem for critical marked Hawkes processes

We prove a central limit type theorem for critical marked Hawkes processes. We study the case where the marks are i.i.d. with nonnegative values and their common distribution is either heavy tailed or has finite variance. The kernel function is of a multiplicative form and the mean number of future events triggered by a single event is $1$ (criticality). We also assume that the base intensity function is heavy tailed. We prove convergence in law in the space of tempered distributions of the normalized empirical measure corresponding to the times of events. We also study convergence in law in the Skorokhod space of the normalized event counting process as the time is speeded up. In case when the distribution of marks is heavy tailed, the limit process is a stable process with dependent increments, while in case of finite variance, the limit process is the same Gaussian process as for the non marked Hawkes process. We develop a new, robust method that may be applied to other self-exciting systems generalizing Hawkes processes. For example we consider a non marked self-exciting system where the number of excitations caused by single event is heavy tailed.

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arXiv2024

On small deviations of Gaussian multiplicative chaos with a strictly logarithmic covariance on Euclidean ball

Recognizing the regime of positive definiteness for a strictly logarithmic covariance kernel, we prove that the small deviations of a related Gaussian multiplicative chaos (GMC) $M_\gamma$ are for each natural dimension $d$ always of lognormal type, i.e. the upper and lower limits as $t\to \infty$ of $$ -\ln\Big(\mathbb{P}(M_\gamma(B(0,r))\le \delta \Big)/(\ln \delta)^2 $$ are finite and bounded away from zero. We then place the small deviations in the context of Laplace transforms of $M_\gamma$ and discuss the explicit bounds on the associated constants. We also provide some new representations of the Laplace transform of GMC related to a strictly logarithmic covariance kernel.

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arXiv2020

Time regularity of L\'{e}vy-type evolution in Hilbert spaces and of some $\alpha$-stable processes

In this paper we consider the existence of weakly c\`adl\`ag versions of a solution to a linear equation in a Hilbert space $H$, driven by a Levy process taking values in a Hilbert space $U$. In particular we are interested in diagonal type processes, where process on coordinates are functionals of independent $\alpha$ stable symmetric process. We give the if and only if characterization in this case. We apply the same techniques to obtain a sufficient condition for existence of a c\`adl\`ag versions of stable processes described as integrals of deterministic functions with respect to symmetric $\alpha$-stable random measures with $\alpha\in[1,2)$.

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arXiv2019

Limit theorems for integrated trawl processes with symmetric L\'evy bases

We study long time behavior of integrated trawl processes introduced by Barndorff-Nielsen. The trawl processes form a class of stationary infinitely divisible processes, described by an infinitely divisible random measure (L\'evy base) and a family of shifts of a fixed set (trawl). We assume that the L\'evy base is symmetric and homogeneous and that the trawl set is determined by the trawl function that decays slowly. Depending on the geometry of the trawl set and on the L\'evy measure corresponding to the L\'evy base we obtain various types of limits in law of the normalized integrated trawl processes for large times. The limit processes are always stable and self-similar with stationary increments. In some cases they have independent increments - they are stable L\'evy processes where the index of stability depends on the parameters of the model. We show that stable limits with stability index smaller than 2 may appear even in cases when the underlying L\'evy base has all its moments finite. In other cases, the limit process has dependent increments and it may be considered as a new extension of fractional Brownian motion to the class of stable processes.

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arXiv2019

Bifractional Brownian motion for $H>1$ and $2HK\le 1$

Bifractional Brownian motion on $\mathbb{R}_+$ is a two parameter centered Gaussian process with covariance function: \[ R_{H,K} (t,s)=\frac 1{2^K}\left(\left(t^{2H}+s^{2H}\right)^K-\ |{t-s}\ |^{2HK}\right), \qquad s,t\ge 0. \] This process has been originally introduced by Houdr\'e and Villa (2003) for the range of parameters $H\in (0,1]$ and $K\in (0,1]$. Since then, the range of parameters, for which $R_{H,K}$ is known to be nonnegative definite has been somewhat extended, but the full range is still not known. We give an elementary proof that $R_{H,K}$ is nonnegative definite for parameters $H,K$ satisfying $H>1$ and $0<2HK\le 1$. We show that $R_{H,K}$ can be decomposed into a sum of two nonnegative definite functions. As a side product we obtain a decomposition of the fractional Brownian motion with Hurst parameter $H<\frac 12$ into a sum of time rescaled Brownian motion and another independent self-similar Gaussian process. We also discuss some simple properties of bifractional Brownian motion with $H>1$.

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arXiv2017

Non-Gaussian limit of a tracer motion in an incompressible flow

We consider a massless tracer particle moving in a random, stationary, isotropic and divergence free velocity field. We identify a class of fields, for which the limit of the laws of appropriately scaled tracer trajectory processes is non-Gaussian but a Rosenblatt type of process.

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arXiv2014

Diffusion limits at small times for coalescents with a Kingman component

We consider standard $\La$-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". Equivalently, the driving measure $\Lambda$ has an atom at $0$; $\Lambda(\{0\})=c>0$. It is known that all such coalescents come down from infinity. Moreover, the number of blocks $N_t$ is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we investigate the second-order asymptotics of $N_t$ in the functional sense at small times. This complements our earlier results on the fluctuations of the number of blocks for a class of regular $\La$-coalescents without the Kingman part. In the present setting it turns out that the Kingman part dominates, and the limit process is a Gaussian diffusion, as opposed to the stable limit in our previous work.

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arXiv2013

From intersection local time to the Rosenblatt process

The Rosenblatt process was obtained by Taqqu (1975) from convergence in distribution of partial sums of strongly dependent random variables. In this paper we give a particle picture approach to the Rosenblatt process with the help of intersection local time and white noise analysis, and discuss measuring its long range dependence by means of a number called dependence exponent.

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arXiv2013

Second-order asymptotics for the block counting process in a class of regularly varying $\Lambda$-coalescents

Consider a standard ${\Lambda }$-coalescent that comes down from infinity. Such a coalescent starts from a configuration consisting of infinitely many blocks at time $0$, but its number of blocks $N_t$ is a finite random variable at each positive time $t$. Berestycki et al. [Ann. Probab. 38 (2010) 207-233] found the first-order approximation $v$ for the process $N$ at small times. This is a deterministic function satisfying $N_t/v_t\to1$ as $t\to0$. The present paper reports on the first progress in the study of the second-order asymptotics for $N$ at small times. We show that, if the driving measure $\Lambda$ has a density near zero which behaves as $x^{-\beta}$ with $\beta\in(0,1)$, then the process $(\varepsilon^{-1/(1+\beta)}(N_{\varepsilon t}/v_{\varepsilon t}-1))_{t\ge0}$ converges in law as $\varepsilon\to0$ in the Skorokhod space to a totally skewed $(1+\beta)$-stable process. Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein-Uhlenbeck type, with a completely asymmetric stable L\'{e}vy noise.

math.PR↗

arXiv2012

Oscillatory Fractional Brownian Motion and Hierarchical Random Walks

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the discrete and ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. We also give other results to provide an overall picture of the behavior of this kind of systems, emphasizing the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.

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arXiv2011

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion (fBm), sub-fractional Brownian motion (sub-fBm), negative sub-fractional Brownian motion (nsfBm) and the odd part of fBm in the sense of Dzhaparidze and van Zanten (2004) by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter. Particle picture interpretations of sub-fBm and nsfBm were known earlier (using a different approach) for narrow ranges of parameters; the odd part of fBm process had not been given any physical interpretation at all. Our approach consists in representing these processes as $ $, $ $, $ $, respectively, where X(1) is an (extended) $S'$-random variable obtained as the fluctuation limit of either empirical process or the occupation time process of an appropriate particle system. In fact, our construction is more general, permitting to obtain some new Gaussian processes, as well as multidimensional random fields. In particular, we generalize and presumably simplify some results by Hambly and Jones (2007). We also obtain a new class of $S'$-valued density processes, containing as a particular case the density process of Martin-Löf (1976).

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arXiv2010

Number variance for hierarchical random walks and related fluctuations

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the c-random walks. Such walks are used, e.g., in population genetics. The number variance problem consists in investigating if the variance of the number of "particles" N_n(L) lying in the ball of radius L at a given time n remains bounded, or even better, converges to a finite limit, as $L\to \infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of N_n(L) around the mean as $n\to \infty$ and L is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the c-random walks mimic symmetric stable processes on R, we compare our results to those obtained by Hambly and Jones (2007,2009), where the number variance problem for an infinite system of symmetric stable processes on R was studied. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a centered Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments.

math.PR↗

arXiv2010

Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in "low" dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for "low" dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type.

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arXiv2008

Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses

The $(d,α,β,γ)$-branching particle system consists of particles moving in $R^d$ according to a symmetric $α$-stable Lévy process $(0<α\leq 2)$, splitting with a critical $(1+β)$-branching law $(0<β\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $μ_γ(dx)=dx/(1+|x|^γ), γ\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_Tμ_γ$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($γ=0$, the homogeneous case), and finite measures $(γ>d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $μ_γ, γ\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,α,β,γ$. Related results for the corresponding $(d,α,β,γ)$-superprocess are also given.

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arXiv2007

Self-similar stable processes arising from high-density limits of occupation times of particle systems

We extend results on time-rescaled occupation time fluctuation limits of the $(d,α, β)$-branching particle system $(0<α\leq 2, 0<β\leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>α/ β$ only, since the particle system becomes locally extinct if $d\le α/ β$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>α/β$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<α(1+β)/β$ and $d<α(2+β)/(1+β)$, respectively) the limits are determined by non-Lévy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\cal S}'(R^d)$-valued Lévy processes in the Lebesgue case, stable processes constant in time on $(0,\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $β=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes.

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arXiv2007

A long range dependence stable process and an infinite variance branching system

We prove a functional limit theorem for the rescaled occupation time fluctuations of a $(d,α,β)$-branching particle system [particles moving in $\mathbb {R}^d$ according to a symmetric $α$-stable Lévy process, branching law in the domain of attraction of a $(1+β)$-stable law, $0<β<1$, uniform Poisson initial state] in the case of intermediate dimensions, $α/β d/(d+α)$, which coincides with the case of finite variance branching $(β=1)$, and another one for $β\leq d/(d+α)$, where the long range dependence depends on the value of $β$. The long range dependence is characterized by a dependence exponent $κ$ which describes the asymptotic behavior of the codifference of increments of $ξ$ on intervals far apart, and which is $d/α$ for the first case (and for $α=2$) and $(1+β-d/(d+α))d/α$ for the second one. The convergence proofs use techniques of $\mathcal{S}'(\mathbb {R}^d)$-valued processes.

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arXiv2007

Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance \int_0^{s\wedge t} u^a [(t-u)^b+(s-u)^b]du, parameters a>-1, -1<b\leq 1, |b|\leq 1+a, corresponds to fractional Brownian motion for a=0, -1<b<1. The second one, with covariance (2-h)(s^h+t^h-[(s+t)^h +|s-t|^h]/2), parameter 0<h\leq 4, corresponds to sub-fractional Brownian motion for 0<h<2. The third one, with covariance -(s^2\log s + t^2\log t -[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]/2), is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.

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