arXiv subjects

Dasha Loukianova

Publications and source records attributed to Dasha Loukianova.

At least 19 recordsLinked to original sources

arXiv2026

Weak conditional propagation of chaos for systems of interacting particles with nearly stable jumps

We consider a system of $N$ interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a heavy-tailed random variable belonging to the domain of attraction of an $α$-stable law and scaled by $N^{-1/α},$ where $0<α<2$. We call these jumps collateral jumps. Moreover, in case $0<α<1$, the jumping particle itself undergoes a macroscopic, main jump. Such systems appear in the modeling of large neural networks, such as the human brain. Using a representation of the collateral jump sum as a time-changed random walk, we prove the convergence in law, in Skorokhod space, of this system to a limit infinite-exchangeable system of SDEs driven by a common stable process. This stable process arises due to the stable central limit theorem, and the particles in the limit system are independent and identically distributed, conditionally on that. That is, the $N$-particle system exhibits the conditional propagation of chaos property.

math.PR↗

arXiv2026

Nonparametric estimation of the jump rate in mean field interacting systems of neurons

We consider finite systems of $N$ interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of $N$ neurons over a fixed time interval $[0,t]$. Asymptotic are taken as $N,$ the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an $L^2 -$error loss over a range of Hölder classes and obtain the classical rate of convergence $ N^{ - 2β/ ( 2 β+ 1)} , $ where $ β$ is the regularity of the unknown spiking rate function.

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arXiv2026

LAN property for the parameter of the jump rate in mean field interacting systems of neurons

In the context of a large system of $N$ neurons interacting through spike events in a mean-field regime as $N\to \infty$, we characterize the estimation of a multidimensional parameter in the spiking rate, when the neural states are observed over a fixed time horizon. We first prove the local asymptotic normality (LAN) property and leverage classical theory to establish the asymptotic efficiency of the maximum likelihood estimator. While the theory of Ibragimov and Hasminskii yields strong results, up to global asymptotic minimax bound, its applicability appears currently limited to models without state resets at spike times. Following then Höpfner's classical approach, we nevertheless derive, in a general setting including neuron reset, the consistency, asymptotic normality and local asymptotic minimax optimality of the estimator. Keywords: Local Asymptotic Normality (LAN); Mean-field regime; Interacting particle system; Multidimensional parameter estimation; Jump rate estimation; Maximum likelihood estimator (MLE); Asymptotic minimax optimality

math.ST↗

arXiv2025

Strong propagation of chaos for systems of interacting particles with nearly stable jumps

We consider a system of $N$ interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a heavy tailed random variable belonging to the domain of attraction of an $α-$ stable law and scaled by $N^{-1/α},$ where $0 < α<2 .$ We call these jumps collateral jumps. Moreover, in case $ 0 < α< 1, $ the jumping particle itself undergoes a macroscopic, main jump. Such systems appear in the modeling of large neural networks, such as the human brain. The particular scaling of the collateral jumps implies that the limit of the empirical measures of the system is random and equals the conditional distribution of one typical particle in the limit system, given the source of common noise. Thus the system exhibits the conditional propagation of chaos property. The limit system turns out to be solution of a non-linear SDE, driven by an $ α-$stable process. We prove strong unique existence of the limit system and introduce a suitable coupling to obtain the strong convergence of the finite to the limit system, together with precise error bounds for finite time marginals.

math.PR↗

arXiv2024

Mean field limits of interacting particle systems with positive stable jumps

This note is a companion article to the recent paper Löcherbach, Loukianova, Marini (2024). We consider mean field systems of interacting particles. Each particle jumps with a jump rate depending on its position. When jumping, a macroscopic quantity is added to its own position. Moreover, simultaneously, all other particles of the system receive a small random kick which is distributed according to a positive $α-$stable law and scaled in $N^{-1/α},$ where $0 < α< 1.$ In between successive jumps of the system, the particles follow a deterministic flow with drift depending on their position and on the empirical measure of the total system. In a more general framework where jumps and state space do not need to be positive, we have shown in Löcherbach, Loukianova, Marini (2024) that the mean field limit of this system is a McKean-Vlasov type process which is solution of a non-linear SDE, driven by an $ α-$stable process. Moreover we have obtained in Löcherbach, Loukianova, Marini (2024) an upper bound for the strong rate of convergence with respect to some specific distance disregarding big jumps of the limit stable process. In the present note we consider the specific situation where all jumps are positive and particles take values in $[ 0, + \infty [ . $ We show that in this case it is possible to improve upon the error bounds obtained in Löcherbach, Loukianova, Marini (2024) by using an adhoc distance obtained after applying a concave space transform to the trajectories. The distance we propose here takes into account the big jumps of the limit $ α-$stable subordinator.

math.PR↗

arXiv2022

Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling

We consider the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and Löcherbach (2016) and then further studied in Erny, Löcherbach and Loukianova (2021) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. In a previous article we proved the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in Komlós, Major and Tusnády (1976) of the point process representing the small jumps of the particle system with the limit Brownian motion

math.PR↗

arXiv2022

Limit of the environment viewed from Sinaï's walk

For Sinaï's walk (X_k) we show that the empirical measure of the environment seen from the particle (\bar\w_k) converges in law to some random measure S. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of (\bar\w_k) . When the limit in this "in law" ergodic theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums sum_{k=1}^nf(ΔX_k), where f is bounded and depending on the steps ΔX_k:=X_{k+1}-X_k.

math.PR↗

arXiv2021

White-noise driven conditional McKean-Vlasov limits for systems of particles with simultaneous and random jumps

We study the convergence of $N-$particle systems described by SDEs driven by Brownian motion and Poisson random measure, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position and on the empirical measure of the system. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a random jump height that is centred and scaled in $N^{-1/2}.$ This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditional distribution of one particle in the limit system. We call such limits {\it conditional McKean-Vlasov limits}. The conditioning in the limit measure reflects the dependencies between coexisting particles in the limit system such that we are dealing with a {\it conditional propagation of chaos property}. As a consequence of the scaling in $N^{-1/2}$ and of the fact that the limit of the empirical measures is not deterministic the limit system turns out to be solution of a non-linear SDE, where not independent martingale measures and white noises appear having an intensity that depends on the conditional law of the process.

math.PR↗

arXiv2020

Mean field limits for interacting Hawkes processes in a diffusive regime

We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups.

math.PR↗

arXiv2020

Conditional propagation of chaos for mean field systems of interacting neurons

We study the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and Löcherbach (2016) in a diffusive scaling. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion $W$ which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system. Conditionally on $W,$ the different neurons are independent in the limit system. This is the {\it conditional propagation of chaos} property. We prove the well-posedness of the limit equation by adapting the ideas of Graham (1992) to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

math.PR↗

arXiv2016

Jump filtering and efficient drift estimation for Lévy-driven SDE's

The problem of drift estimation for the solution $X$ of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density these conditions reduce to $n Δ_n^{3-ε}\to 0,$ where $n$ is the number of observations and $Δ_n$ is the maximal sampling step. This result relaxes the condition $nΔ_n^2 \to 0$ usually required for joint estimation of drift and diffusion coefficient for SDE's with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part $X^c$ in the likelihood function. In order to construct the drift estimator we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to $X^c$. Convergence results of independent interest are proved for these nonparametric estimators. Finally, we illustrate the behavior of our drift estimator for a number of popular Lévy-driven models from finance.

math.ST↗

arXiv2015

Parametric estimation of a one-dimensional ballistic random walk in a Markov environment

We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a one-dimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.

math.ST↗

arXiv2014

Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment

We consider a one dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. In that purpose, we adapt the method developed in the ballistic case by Comets et al (2014) and Falconnet, Loukianova and Matias (2014). Using a supplementary assumption due to the specificity of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphazis the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike in the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.

math.PR↗

arXiv2014

Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support

We consider a one-dimensional recurrent random walk in random environment (RWRE) when the environment is i.i.d. with a parametric, finitely supported distribution. Based on a single observation of the path, we provide a maximum likelihood estimation procedure of the parameters of the environment. Unlike most of the classical maximum likelihood approach, the limit of the criterion function is in general a nondegenerate random variable and convergence does not hold in probability. Not only the leading term but also the second order asymptotics is needed to fully identify the unknown parameter. We present different frameworks to illustrate these facts. We also explore the numerical performance of our estimation procedure.

math.PR↗

arXiv2013

Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment

We consider a one dimensional ballistic random walk evolving in a parametric independent and identically distributed random environment. We study the asymptotic properties of the maximum likelihood estimator of the parameter based on a single observation of the path till the time it reaches a distant site. We prove an asymptotic normality result for this consistent estimator as the distant site tends to infinity and establish that it achieves the Cramér-Rao bound. We also explore in a simulation setting the numerical behaviour of asymptotic confidence regions for the parameter value.

math.ST↗

arXiv2012

Maximum likelihood estimator consistency for ballistic random walk in a parametric random environment

We consider a one dimensional ballistic random walk evolving in an i.i.d. parametric random environment. We provide a maximum likelihood estimation procedure of the environment parameters based on a single observation of the path till the time it reaches a distant site, and prove that this estimator is consistent as the distant site tends to infinity. We also explore the numerical performances of our estimation procedure.

math.ST↗

arXiv2012

Spectral condition, hitting times and Nash inequality

Let $X$ be a $μ$-symmetric Hunt process on a LCCB space E. For an open set G $\subseteq$ E, let $τ_G$ be the exit time of $X$ from G and $A^G$ be the generator of the process killed when it leaves G. Let $r:[0,\infty[\to[0,\infty[$ and $R (t) = \int_0^t r(s) ds$. We give necessary and sufficient conditions for $\E_μ R (τ_G)<\infty$ in terms of the behavior near the origin of the spectral measure of $-A^G.$ When $r(t)=t^l$, $l>0$, by means of this condition we derive the Nash inequality for the killed process. In the case of one-dimensional diffusions, this permits to show that the existence of moments of order $l$ for $τ_G$ implies the Nash inequality of order $p=\frac{l+2}{l+1}$ for the whole process. The associated rate of convergence of the semi-group in $L^2(μ)$ is bounded by $t^{-(l+1)}$. For diffusions in dimension greater than one, we obtain the Nash inequality of the same order under an additional non-degeneracy condition (local Poincaré inequality). Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of order $t^{-(l+1)}$ of the semi-group, implies the existence of moments of order $l+1 -ε$ for $τ_G$, for all $ ε>0$.

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arXiv2011

Polynomial deviation bounds for recurrent Harris processes having general state space

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc, Fort and Guillin (2009) introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the regeneration method of Nummelin, extended to the time-continuous context. As a consequence, if a $p-$th moment of the regeneration times exists, we obtain non asymptotic deviation bounds of the form $$P_ν(|\frac1t\int_0^tf(X_s)ds-μ(f)|\geq\ge)\leq K(p)\frac1{t^{p- 1}}\frac 1{\ge^{2(p-1)}}\|f\|_\infty^{2(p-1)}, p \geq 2. $$ Here, $f$ is a bounded function and $μ$ is the invariant measure of the process. We give several examples, including elliptic stochastic differential equations and stochastic differential equations driven by a jump noise.

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