Recently a functional limit theorem for sums of moving averages with random coefficients and i.i.d. heavy tailed innovations has been obtained under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes place in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology. In this article we extend this result to the case when the innovations are weakly dependent in the sense of strong mixing and local dependence condition $D'$.
We investigate maxima of linear processes with i.i.d. heavy-tailed innovations and random coefficients. Using the point process approach we derive functional convergence of the partial maxima stochastic process in the space of non-decreasing càdlàg functions on $[0,1]$ with the Skorohod $M_{1}$ topology.
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $\mathbb{R}^{2}$-valued càdlàg functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients necessary and sufficient conditions for the finite dimensional convergence to an $α$-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $α\leq 1$. In the functional setting we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linea…
We look at joint regular variation properties of MA($\infty$) processes of the form $\mathbf{X} = (X_k, k \in \mathbb{Z})$ where $X_k = \sum_{j=0}^{\infty} ψ_j Z_{k-j}$ and the sequence of random variables $(Z_i, i \in \mathbb{Z})$ are i.i.d. with regularly varying tails. We use the setup of $\mathbb{M}_{\mathbb{O}}$-convergence and obtain hidden regular variation properties for $\mathbf{X}$ under suitable summability conditions on the constant coefficients $(ψ_j : j \geq 0)$. Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.
In this paper we extend the refined second-order Poincaré inequality for Poisson functionals from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin-Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavy-tailed moving averages. The extension allows a functional with multivariate arguments, i.e. multiple moving averages and also multivariate values of the functional. Such a set-up has previously not been explored in the framework of stable moving av…
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $α$-stable limit distributions o…
In this paper we obtain Berry-Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $α$, and its tail-index, which is controlled by a parameter $β$. In fact, we obtain the classical $\sqrt{1/ n}$ rate of convergence when the tails are not too heavy and the memory is not too stro…
Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of an ellipticity condition. More precisely, we classify these systems according to their type and --- in the recurrent case --- provide with sharp conditions quantifying the nature of recurrence by establishing which moments of passage times exist and which do not exist. The problem is tackled by mapping the random dynamical systems…
We investigate how large deviations events cluster in the framework of an infinite moving average process with light-tailed noise and long memory. The long memory makes clusters larger, and the asymptotic behaviour of the size of the cluster turns out to be described by the first hitting time of a randomly shifted fractional Brownian motion with drift.
We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we obtain functional convergence of the corresponding partial sum stochastic process in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology.
In the paper we consider the partial sum process $\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}=\sum_{j=0}^{\infty} a_{j}^{(n)}ξ_{k-j}(b(n)), \ k\in \bz\},\ n\ge 1,$ is a series of linear processes with tapered filter $a_{j}^{(n)}=a_{j}\ind{[0\le j\le ł(n)]}$ and heavy-tailed tapered innovations $ξ_{j}(b(n), \ j\in \bz$. Both tapering parameters $b(n)$ and $ł(n)$ grow to $\infty$ as $n\to \infty$. The limit behavior of the partial sum process depends on the growth of these two tapering parameters and dependence properties of a linear process with non-tapered filter $a_i, \ i\ge 0$ and non-ta…
In the paper we consider the partial sum process $\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}=\sum_{j=0}^{\infty} a_{j}^{(n)}ξ_{k-j}(b(n)), \ k\in \bz\},\ n\ge 1,$ is a series of linear processes with tapered filter $a_{j}^{(n)}=a_{j}\ind{[0\le j\le ł(n)]}$ and heavy-tailed tapered innovations $ξ_{j}(b(n), \ j\in \bz$. Both tapering parameters $b(n)$ and $ł(n)$ grow to $\infty$ as $n\to \infty$. The limit behavior of the partial sum process (in the sense of convergence of finite dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a…
Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process…
Consider a branching system with particles moving according to an Ornstein-Uhlenbeck process with drift $μ>0$ and branching according to a law in the domain of attraction of the $(1+β)$-stable distribution. The mean of the branching law is strictly larger than $1$ implying that the system is supercritical and the total number of particles grows exponentially at some rate $λ>0$. It is known that the system obeys a law of large numbers. In the paper we study its rate of convergence. We discover an interesting interplay between the branching rate $λ$ and the drift parameter $μ$. There are three r…
We investigate long and short memory in $α$-stable moving averages and max-stable processes with $α$-Fréchet marginal distributions. As these processes are heavy-tailed, we rely on the notion of long range dependence suggested by Kulik and Spodarev (2019) based on the covariance of excursions. Sufficient conditions for the long and short range dependence of $α$-stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max-stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependen…
We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random…
Marcinkiewicz strong law of large numbers, ${n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\ $ almost surely with $p\in(1,2)$, are developed for products $d_k=\prod_{r=1}^s x_k^{(r)}$, where the $x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}ξ_l^{(r)}$ are two-sided linear processes with coefficients $\{c_l^{(r)}\}_{l\in \mathbb{Z}}$ and i.i.d. zero-mean innovations $\{ξ_l^{(r)}\}_{l\in \mathbb{Z}}$. The decay of the coefficients $c_l^{(r)}$ as $|l|\to\infty$, can be slow enough for $\{x_k^{(r)}\}$ to have long memory while $\{d_k\}$ can have heavy tails. The long-range dependence and…
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.
Central limit theory for serial tail dependence estimators in heavy-tailed long memory linear time series
University of Stuttgart（斯图加特大学）；Stuttgart Center for Simulation Science (SC SimTech), University of Stuttgart（斯图加特大学斯图加特模拟科学中心）；Univ Angers, CNRS, LAREMA, SFR MATHSTIC（昂热大学、法国国家科学研究中心、LAREMA、MATHSTIC研究联盟）
Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented.
We present an outline of the theory of certain Lévy-driven, multivariate stochastic processes, where the processes are represented by rational transfer functions (Continuous-time AutoRegressive Moving Average or CARMA models) and their applications in non-Gaussian time series modelling. We discuss in detail their definition, their spectral representation, the equivalence to linear state space models and further properties like the second order structure and the tail behaviour under a heavy-tailed input. Furthermore, we study the estimation of the parameters using quasi-maximum likelihood estim…
