A family of nonparametric unit root tests for processes driven by infinite variance innovations

This paper presents extensions to the family of nonparametric fractional variance ratio (FVR) unit root tests of Nielsen (2009. “A Powerful Test of the Autoregressive Unit Root Hypothesis Based on a Tuning Parameter Free Statistic.” Econometric Theory 25: 1515–44) under heavy tailed (infinite variance) innovations. In this regard, we first develop the asymptotic theory for these FVR tests under this setup. We show that the limiting distributions of the tests are free of serial correlation nuisance parameters, but depend on the tail index of the infinite variance process. Then, we compare the finite sample size and power performance of our FVR unit root tests with the well-known parametric ADF test under the impact of the heavy tailed shocks. Simulations demonstrate that under heavy tailed innovations, the nonparametric FVR tests have desirable size and power properties.

Asymptotic behavior of dependence measures for Ornstein-Uhlenbeck model based on long memory processes

In this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model. Their are extensions of the Ornstein-Uhlenbeck system for which in the classic version we replace the standard Brownian motion (or other L$$\acute{e}$$ e ´ vy process) by long range dependent processes based on $$\alpha -$$ α - stable distribution. One way of characterizing long- and short-range dependence of second order processes is in terms of autocovariance function. However, for systems with infinite variance the classic measure is not defined, therefore there is a need to consider alternative measures on the basis of which the long range dependence can be recognized. In this paper, we study three alternative measures adequate for $$\alpha -$$ α - stable-based processes. We calculate them for examined processes and indicate their asymptotic behavior. We show that one of the analyzed Ornstein-Uhlenbeck process exhibits long memory property while the second does not. Moreover, we show the ratio of two introduced measures is limited which can be a starting point to introduction of a new estimation method of stability index for analyzed Ornstein-Uhlenbeck processes.

On Estimation of the Convergence Rate to Invariant Measures in Markov Branching Processes with Possibly Infinite Variance and Allowing Immigration

The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming that the nonlinear parts of the appropriate generating functions are regularly varying in the sense of Karamata, we prove theorems on convergence of transition functions of the process to invariant measures. We deduce the speed rate of these convergence providing that slowly varying factors are with remainder

Gamma-ray blazar variability: new statistical methods of time-flux distributions

ABSTRACT Variable γ-ray emission from blazars, one of the most powerful classes of astronomical sources featuring relativistic jets, is a widely discussed topic. In this work, we present the results of a variability study of a sample of 20 blazars using γ-ray (0.1–300 GeV) observations from Fermi/LAT telescope. Using maximum likelihood estimation (MLE) methods, we find that the probability density functions that best describe the γ-ray blazar flux distributions use the stable distribution family, which generalizes the Gaussian distribution. The results suggest that the average behaviour of the γ-ray flux variability over this period can be characterized by log-stable distributions. For most of the sample sources, this estimate leads to standard lognormal distribution (α = 2). However, a few sources clearly display heavy tail distributions (MLE leads to α < 2), suggesting underlying multiplicative processes of infinite variance. Furthermore, the light curves were analysed by employing novel non-stationarity and autocorrelation analyses. The former analysis allowed us to quantitatively evaluate non-stationarity in each source – finding the forgetting rate (corresponding to decay time) maximizing the log-likelihood for the modelled evolution of the probability density functions. Additionally, evaluation of local variability allows us to detect local anomalies, suggesting a transient nature of some of the statistical properties of the light curves. With the autocorrelation analysis, we examined the lag dependence of the statistical behaviour of all the {(yt, yt + l)} points, described by various mixed moments, allowing us to quantitatively evaluate multiple characteristic time scales and implying possible hidden periodic processes.

Long range dependence of heavy-tailed random functions

We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

Non-Gaussian fluctuations of randomly trapped random walks

In this paper we consider the one-dimensional, biased, randomly trapped random walk with infinite-variance trapping times. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton–Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

Quantile inference for nonstationary processes with infinite variance innovations

Based on the quantile regression, we extend Koenker and Xiao (2004) and Ling and McAleer (2004)’s works from finite-variance innovations to infinite-variance innovations. A robust t-ratio statistic to test for unit-root and a re-sampling method to approximate the critical values of the t-ratio statistic are proposed in this paper. It is shown that the limit distribution of the statistic is a functional of stable processes and a Brownian bridge. The finite sample studies show that the proposed t-ratio test always performs significantly better than the conventional unit-root tests based on least squares procedure, such as the Augmented Dick Fuller (ADF) and Philliphs-Perron (PP) test, in the sense of power and size when infinite-variance disturbances exist. Also, quantile Kolmogorov-Smirnov (QKS) statistic and quantile Cramer-von Mises (QCM) statistic are considered, but the finite sample studies show that they perform poor in power and size, respectively. An application to the Consumer Price Index for nine countries is also presented.

On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance

UDC 519.218.2 We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results.