arXiv subjects

Donatas Surgailis

Publications and source records attributed to Donatas Surgailis.

At least 19 recordsLinked to original sources

arXiv2026

Limit distribution of the sample volume fraction of Boolean set

We study the limit distribution of the volume fraction estimator $\widehat p_{λ, A}$ (= the Lebesgue measure of the intersection $\mathcal{X}\cap (λA)$ of a random set $\mathcal{X}$ with a large observation set $λA$, divided by the Lebesgue measure of $λA$), as $λ\to \infty$, for a Boolean set $\mathcal{X}$ formed by uniformly scattered random grains $Ξ\subset \mathbb{R}^ν$. We obtain general conditions on generic grain set $Ξ$ under which $\widehat p_{λ, A}$ has an $α$-stable limit distribution with index $1 < α\le 2$. A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set observed on a large subset of a $ν_0$-dimensional $(1 \le ν_0 \le ν-1$) hyperplane of $\mathbb{R}^ν$.

math.PR↗

arXiv2023

Scaling limits of nonlinear functions of random grain model, with application to Burgers' equation

We study scaling limits of nonlinear functions $G$ of random grain model $X$ on $\mathbb{R}^d $ with long-range dependence and marginal Poisson distribution. Following Kaj et al (2007) we assume that the intensity $M$ of the underlying Poisson process of grains increases together with the scaling parameter $λ$ as $M = λ^γ$, for some $γ> 0$. The results are applicable to the Boolean model and exponential $G$ and rely on an expansion of $G$ in Charlier polynomials and a generalization of Mehler's formula. Application to solution of Burgers' equation with initial aggregated random grain data is discussed.

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arXiv2023

Scaling transition for singular linear random fields on $\mathbb{Z}^2$: spectral approach

We study partial sums limits of linear random fields $X$ on $\mathbb{Z}^2 $ with spectral density $f({\mathbf x}) $ tending to $\infty,\, 0$ or to both (along different subsequences) as ${\mathbf x} \to (0,0)$. The above behaviors are termed (spectrum) long-range dependence, negative dependence, and long-range negative dependence, respectively, and assume an anisotropic power-law form of $f({\mathbf x})$ near the origin. The partial sums are taken over rectangles whose sides increase as $λ$ and $λ^γ$, for any fixed $γ>0$. We prove that for above $X$ the partial sums or scaling limits exist for any $γ>0$ and exhibit a scaling transition at some $γ= γ_0>0$; moreover, the `unbalanced' scaling limits ($γ\neγ_0$) are Fractional Brownian Sheet with Hurst parameters taking values from $[0,1]$. The paper extends \cite{ps2015, pils2017, sur2020} to the above spectrum dependence conditions and/or more general values of Hurst parameters.

math.PR↗

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

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↗

arXiv2020

Scaling transition and edge effects for negatively dependent linear random fields on ${\mathbb{Z}}^2$

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for a class of negatively dependent linear random fields on ${\mathbb{Z}}^2$ with moving-average coefficients $a(t,s)$ decaying as $|t|^{-q_1}$ and $|s|^{-q_2}$ in the horizontal and vertical directions, $q_1^{-1} + q_2^{-1} < 1 $. The scaling limits are taken over rectangles whose sides increase as $λ$ and $λ^γ$ when $λ\to \infty$, for any $γ>0$. We prove that the scaling transition %and the structure of the scaling diagram in this model is closely related to the presence or absence of the edge effects.

math.PR↗

arXiv2019

Sample covariances of random-coefficient AR(1) panel model

The present paper obtains a complete description of the limit distributions of sample covariances in N x n panel data when N and n jointly increase, possibly at different rate. The panel is formed by N independent samples of length n from random-coefficient AR(1) process with the tail distribution function of the random coefficient regularly varying at the unit root with exponent $β$ > 0. We show that for $β$ $\in$ (0, 2) the sample covariances may display a variety of stable and non-stable limit behaviors with stability parameter depending on $β$ and the mutual increase rate of N and n.

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↗

arXiv2019

Estimating long memory in panel random-coefficient AR(1) data

It is well-known that random-coefficient AR(1) process can have long memory depending on the index $β$ of the tail distribution function of the random coefficient, if it is a regularly varying function at unity. We discuss estimation of $β$ from panel data comprising N random-coefficient AR(1) series, each of length T. The estimator of $β$ is constructed as a version of the tail index estimator of Goldie and Smith (1987) applied to sample lag 1 autocorrelations of individual time series. Its asymptotic normality is derived under certain conditions on N, T and some parameters of our statistical model. Based on this result, we construct a statistical procedure to test if the panel random-coefficient AR(1) data exhibit long memory. A simulation study illustrates finite-sample performance of the introduced estimator and testing procedure.

math.ST↗

arXiv2018

Anisotropic scaling limits of long-range dependent linear random fields on ${\mathbb {Z}}^3$

We provide a complete description of anisotropic scaling limits of stationary linear random field on ${\mathbb {Z}}^3$ with long-range dependence and moving average coefficients decaying as $O(|t_i|^{-q_i})$ in the $i$th direction, $i=1,2,3.$ The scaling limits are taken over rectangles in ${\mathbb {Z}}^3$ whose sides increase as $O(λ^{γ_i}), i=1,2,3$ when $λ\to \infty$, for any fixed $γ_i >0, i=1,2,3 $. We prove that all these limits are Gaussian RFs whose covariance structure essentially is determined by the fulfillment or violation of the balance conditions $γ_i q_i = γ_j q_j, 1 \le i < j \le 3$. The paper extends recent results in \cite{ps2015}, \cite{ps2016}, \cite{pils2016}, \cite{pils2017} on anisotropic scaling of long-range dependent random fields from dimension 2 to dimension 3.

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arXiv2017

Invariance Principles for Tempered Fractionally Integrated Processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in $α$-stable $(1< α\le 2)$ i.i.d. innovations and related tempered linear processes with vanishing tempering parameter $λ\sim λ_*/N$. We show that the limit of the partial sums process takes a different form in the weakly tempered ($λ_* = 0$), strongly tempered ($λ_* = \infty$), and moderately tempered ($0<λ_* < \infty$) cases. These results are used to derive the limit distribution of the OLS estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.

math.PR↗

arXiv2017

Tempered fractional Brownian and stable motions of second kind

Meerschaert and Sabzikar [12], [13] introduced tempered fractional Brownian/stable motion (TFBM/TFSM) by including an exponential tempering factor in the moving average representation of FBM/FSM. The present paper discusses another tempered version of FBM/FSM, termed tempered fractional Brownian/stable motion of second kind (TFBM II/TFSM II).We prove that TFBM/TFSM and TFBM II/TFSM II are different processes. Particularly, large time properties of TFBM II/TFSM II are similar to those of FBM/FSM and are in deep contrast to large time properties of TFBM/TFSM.

math.PR↗

arXiv2017

The Discrete and Continuum Broken Line Process

In this work we introduce the discrete-space broken line process (with discrete and continues parameter values) and derive some of its properties. We explore polygonal Markov fields techniques developed by Arak-Surgailis. The discrete version is presented first and a natural continuum generalization to a continuous object living on the discrete lattice is then proposed and studied. The broken lines also resemble the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and self-dual. For such distributions there is a law of large numbers and the process extends to the infinite lattice. A proof of Burke's theorem emerges from the construction. We present a simple proof of the explicit law of large numbers for last passage percolation as an application. Finally we show that the exponential and geometric distributions are the only non-trivial ones that yield self-duality.

math.PR↗

arXiv2016

Discrete-time trawl processes with long memory

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent `seed' processes on shrinking intervals (`trawl heights'). Related trawl processes in continuous time were studied in Barndorff-Nielsen (2011) and Barndorff-Nielsen et al. (2014), however in our case, the i.i.d. seed processes can be very general and need not be infinitely divisible. In the case when the trawl height decays with the lag as $j^{-α}$ for some $1< α< 2 $, the trawl process exhibits long memory and its covariance decays as $j^{1-α}$. We show that under general conditions on generic seed process, the normalized partial sums of such trawl process may tend either to a fractional Brownian motion or to an $α$-stable Lévy process.

math.PR↗

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

Aggregation of autoregressive random fields and anisotropic long-range dependence

We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields $Y$ on $\mathbb {Z}^2$ whose normalized partial sums on rectangles with sides growing at rates $O(n)$ and $O(n^γ)$ tend to an operator scaling random field $V_γ$ on $\mathbb {R}^2$, for any $γ>0$. The scaling transition is characterized by the fact that there exists a unique $γ_0>0$ such that the scaling limits $V_γ$ are different and do not depend on $γ$ for $γ>γ_0$ and $γ<γ_0$. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of $α$-stable $(1<α\le2)$ aggregated nearest-neighbor autoregressive random fields on $\mathbb{Z}^2$ with a scalar random coefficient $A$ having a regularly varying probability density near the "unit root" $A=1$.

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↗