A Test for Nonlinearity of Time Series with Infinite Variance

A heavy tailed time series that can be represented as an infinite moving average has the property that the sample autocorrelation function (ACF) at lag h converges in probability to a constant (h), although the mathematical correlation typically does not exist. For many nonlinear heavy tailed models, however, the sample ACF at lag h converges in distribution to a nondegenerate random variable. In this paper, a test for (non)linearity of a given infinite variance time series is constructed, based on subsample stability of the sample ACF. The test is applied to several real and simulated datasets.     

Weak convergence with random indices

Suppose {X_nn?-0} are random variables such that for normalizing constants a_n>0, b_n, n?0 we have Y_n(·)=(X[_{n, ·}]-b_n/a_n ? Y(·) in D(0.∞) . Then a_n and b_n must in specific ways and the process Y possesses a scaling property. If {N_n} are positive integer valued random variables we discuss when Y_Nn → Y and Y'_n=(X[_Nn]-b_n)/a_n ? Y'. Results given subsume random index limit theorems for convergence to Brownian motion, stable processes and extremal processes.     

A bivariate stable characterization and domains of attraction

Bivariate stable distributions are defined as those having a domain of attraction, where vectors are used for normalization. These distributions are identified and their domains of attraction are given in a number of equivalent forms. In one case, marginal convergence implies joint convergence. A bivariate optional stopping property is given. Applications to bivariate random walk are suggested.     

More limit theory for the sample correlation function of moving averages

Let X_t = Σ^∞_j=-∞ c_jZ_{t - j} be a moving average process where {Z_t} is iid with common distribution in the domain of attraction of a stable law with index , 0 < < 2. If 0 < < 2, E|Z₁| < ∞ and the distribution of |Z₁|and |Z₁Z₂| are tail equivalent then the sample correlation function of {X₁} suitably normalized converges in distribution to the ratio of two dependent stable random variables with indices and /2. This is in sharp contrast to the case E|Z₁| ^{= ∞} where the limit distribution is that of the ratio of two independent stable variables. Proofs rely heavily on point process techniques. We also consider the case when the sample correlations are asymptotically normal and extend slightly the classical result.     

Synthetic control over magnetic moment and exchange bias in all-oxide materials encapsulated within a spherical protein cage

This work focuses on the synthetic control of magnetic properties of mixed oxide magnetic nanoparticles of the general formula Fe(3-x)Co(x)O(4) (x < or = 0.33) in the protein cage ferritin. In this biomimetic approach, variations in the chemical synthesis result in the formation of single-phase Fe(3-x)Co(x)O(4) alloys or intimately mixed binary phase Fe/Co oxides, modifying the chemical structure and magnetic behavior of these particles, as characterized by static and dynamic magnetization measurements and X-ray absorption spectroscopy.     

Hidden regular variation under full and strong asymptotic dependence

Data exhibiting heavy-tails in one or more dimensions is often studied using the framework of regular variation. In a multivariate setting this requires identifying specific forms of dependence in the data; this means identifying that the data tends to concentrate along particular directions and does not cover the full space. This is observed in various data sets from finance, insurance, network traffic, social networks, etc. In this paper we discuss the notions of full and strong asymptotic dependence for bivariate data along with the idea of hidden regular variation in these cases. In a risk analysis setting, this leads to improved risk estimation accuracy when regular methods provide a zero estimate of risk. Analyses of both real and simulated data sets illustrate concepts of generation and detection of such models.     

Smoothing the Moment Estimator of the Extreme Value Parameter

Let {X _n be a sequence of i.i.d. random variables whose common distribution F belongs to the domain of attraction of an extreme value law. A semi-parametric estimator of the extreme value parameter is the Dekkers, Einmahl and de Haan [8] moment estimator. Practical use of this estimator requires the problematic choice of a number k=k(n) of upper order statistics and there are few reliable guidelines for this choice. An averaging or smoothing technique is proposed for this estimator yielding a less volatile function of k which in practice aids estimation.     

Pitfalls of Fitting Autoregressive Models for Heavy-Tailed Time Series

We consider the analysis of time series data which require models with a heavy-tailed marginal distribution. A natural model to attempt to fit to time series data is an autoregression of order p, where p itself is often determined from the data. Several methods of parameter estimation for heavy tailed autoregressions have been considered, including Yule–Walker estimation, linear programming estimators, and periodogram based estimators. We investigate the statistical pitfalls of the first two methods when the models are mis-specified—either completely or due to the presence of outliers. We illustrate the results of our considerations on both simulated and real data sets. A warning is sounded against the assumption that autoregressions will be an applicable class of models for fitting heavy tailed data.