arXiv subjects

Vytautė Pilipauskaitė

Publications and source records attributed to Vytautė Pilipauskaitė.

12 recordsLinked to original sources

arXiv2026

Kinetic interacting particle system: parameter estimation from complete and partial discrete observations

In this paper, we study the estimation of drift and diffusion coefficients in a two dimensional system of N interacting particles modeled by a degenerate stochastic differential equation. We consider both complete and partial observation cases over a fixed time horizon [0, T] and propose novel contrast functions for parameter estimation. In the partial observation scenario, we tackle the challenge posed by unobserved velocities by introducing a surrogate process based on the increments of the observed positions. This requires a modified contrast function to account for the correlation between successive increments. Our analysis demonstrates that, despite the loss of Markovianity due to the velocity approximation in the partial observation case, the estimators converge to a Gaussian distribution (with a correction factor in the partial observation case). The proofs are based on Ito like bounds and an adaptation of the Euler scheme. Additionally, we provide insights into Hörmander's condition, which helps establish hypoellipticity in our model within the framework of stochastic calculus of variations.

math.ST↗

arXiv2024

Polynomial rates via deconvolution for nonparametric estimation in McKean-Vlasov SDEs

This paper investigates the estimation of the interaction function for a class of McKean-Vlasov stochastic differential equations. The estimation is based on observations of the associated particle system at time $T$, considering the scenario where both the time horizon $T$ and the number of particles $N$ tend to infinity. Our proposed method recovers polynomial rates of convergence for the resulting estimator. This is achieved under the assumption of exponentially decaying tails for the interaction function. Additionally, we conduct a thorough analysis of the transform of the associated invariant density as a complex function, providing essential insights for our main results.

math.ST↗

arXiv2023

Parameter estimation of discretely observed interacting particle systems

In this paper, we consider the problem of joint parameter estimation for drift and diffusion coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval $[0, T]$, we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($Δ_n$) and the number of particles ($N$) satisfy $Δ_n \rightarrow 0$ and $N \rightarrow \infty$, and asymptotically normal when additionally the condition $Δ_n N \rightarrow 0$ holds.

math.ST↗

arXiv2022

Aggregation of network traffic and anisotropic scaling of random fields

We discuss joint spatial-temporal scaling limits of sums $A_{λ,γ}$ (indexed by $(x,y) \in \mathbb{R}^2_+$) of large number $O(λ^γ)$ of independent copies of integrated input process $X = \{X(t), t \in \mathbb{R}\}$ at time scale $λ$, for any given $γ>0$. We consider two classes of inputs $X$: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that normalized random fields $A_{λ,γ}$ tend to an $α$-stable Lévy sheet $(1< α<2)$ if $ γ< γ_0$, and to a fractional Brownian sheet if $γ> γ_0$, for some $γ_0>0$. We also prove an `intermediate' limit for $γ= γ_0$. Our results extend previous works Mikosch et al. (2002), Gaigalas, Kaj (2003) and other papers to more general and new input processes.

math.PR↗

arXiv2022

Scaling limits of linear random fields on ${\mathbb{Z}}^2$ with general dependence axis

We discuss anisotropic scaling of long-range dependent linear random fields $X$ on ${\mathbb{Z}}^2$ with arbitrary dependence axis (direction in the plane along which the moving-average coefficients decay at a smallest rate). The scaling limits are taken over rectangles whose sides are parallel to the coordinate axes and increase as $λ$ and $λ^γ$ when $λ\to \infty$, for any $γ>0$. The scaling transition occurs at $γ^X_0 >0$ if the scaling limits of $X$ are different and do not depend on $γ$ for $γ> γ^X_0 $ and $γ< γ^X_0$. We prove that the fact of `oblique' dependence axis (or incongruous scaling) dramatically changes the scaling transition in the above model so that $γ_0^X = 1$ independently of other parameters, contrasting the results in Pilipauskaitė and Surgailis (2017) on the scaling transition under congruous scaling.

math.PR↗

arXiv2022

Power variations for fractional type infinitely divisible random fields

This paper presents new limit theorems for power variation of fractional type symmetric infinitely divisible random fields. More specifically, the random field $X = (X(\boldsymbol{t}))_{\boldsymbol{t} \in [0,1]^d}$ is defined as an integral of a kernel function $g$ with respect to a symmetric infinitely divisible random measure $L$ and is observed on a grid with mesh size $n^{-1}$. As $n \to \infty$, the first order limits are obtained for power variation statistics constructed from rectangular increments of $X$. The present work is mostly related to Basse-O'Connor, Lachièze-Rey, Podolskij (2017), Basse-O'Connor, Heinrich, Podolskij (2019), who studied a similar problem in the case $d=1$. We will see, however, that the asymptotic theory in the random field setting is much richer compared to Basse-O'Connor, Lachièze-Rey, Podolskij (2017), Basse-O'Connor, Heinrich, Podolskij (2019) as it contains new limits, which depend on the precise structure of the kernel $g$. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional $β$-stable sheet, and the moving average fractional $β$-stable field, and discuss potential consequences for statistical inference.

math.PR↗

arXiv2022

Local scaling limits of Lévy driven fractional random fields

We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields $X$ on ${\mathbb{R}}^2$ written as stochastic integral with respect to infinitely divisible random measure. The scaling procedure involves increments of $X$ over points the distance between which in the horizontal and vertical directions shrinks as $O(λ)$ and $O(λ^γ)$ respectively as $λ\downarrow 0$, for some $γ>0$. We consider two types of increments of $X$: usual increment and rectangular increment, leading to the respective concepts of $γ$-tangent and $γ$-rectangent random fields. We prove that for above $X$ both types of local scaling limits exist for any $γ>0$ and undergo a transition, being independent of $γ>γ_0$ and $γ<γ_0$, for some $γ_0>0$; moreover, the "unbalanced" scaling limits ($γ\neγ_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskaitė and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on ${\mathbb{Z}}^2$ and Benassi et al. (2004) on $1$-tangent limits of isotropic fractional Lévy random fields.

math.PR↗

arXiv2021

Semiparametric estimation of McKean-Vlasov SDEs

In this paper we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.

math.ST↗

arXiv2019

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of random-coefficient AR(1) process driven by i.i.d. innovations in the domain of normal attraction of an $α$-stable distribution, $0< α\le 2$, as both $N$ and the time scale $n$ tend to infinity, possibly at a different rate. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $β> 0$, we show that, for $β< \max (α, 1)$, the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $α$, $β$ and the mutual increase rate of $N$ and $n$. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $α= 2$ to $0 < α< 2$.

math.ST↗

arXiv2016

Nonparametric estimation of the distribution of the autoregressive coefficient from panel random-coefficient AR(1) data

We discuss nonparametric estimation of the distribution function $G(x)$ of the autoregressive coefficient $a \in (-1,1)$ from a panel of $N$ random-coefficient AR(1) data, each of length $n$, by the empirical distribution function of lag 1 sample autocorrelations of individual AR(1) processes. Consistency and asymptotic normality of the empirical distribution function and a class of kernel density estimators is established under some regularity conditions on $G(x)$ as $N$ and $n$ increase to infinity. The Kolmogorov-Smirnov goodness-of-fit test for simple and composite hypotheses of Beta distributed $a$ is discussed. A simulation study for goodness-of-fit testing compares the finite-sample performance of our nonparametric estimator to the performance of its parametric analogue discussed in Beran et al. (2010).

math.ST↗

arXiv2016

Scaling transition for nonlinear random fields with long-range dependence

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for nonlinear functions (Appell polynomials) of stationary linear random fields on $\mathbb{Z}^2$ with moving average coefficients decaying at possibly different rate in the horizontal and vertical direction. The paper extends recent results on scaling transition for linear random fields in Puplinskaitė and Surgailis (2016), Puplinskaitė and Surgailis (2015).

math.PR↗

arXiv2015

Anisotropic scaling of random grain model with application to network traffic

We obtain a complete description of anisotropic scaling limits of random grain model on the plane with heavy tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian and some `intermediate' infinitely divisible random fields. Asymptotic form of the covariance function of the random grain model is obtained. Application to superposed network traffic is included.

math.PR↗