Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Summary. Consider the stationary linear process , , where is an i.i.d. finite variance sequence. The spectral density of may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form , where denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels and for to converge to a Gaussian distribution. We show that this convergence holds if and are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain. Received: 28 February 1996 / In revised form: 10 July 1996     

A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate

Summary A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.     

Asymptotic normality of quadratic forms of martingale differences

We establish the asymptotic normality of a quadratic form \(Q_n\) in martingale difference random variables \(\eta _t\) when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables \(\eta _t\), asymptotic normality holds under condition \(||A||_{sp}=o(||A||) \), where \(||A||_{sp}\) and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences \(\eta _t\) has been an important open problem. We provide such sufficient conditions in this paper.     

ARCH-type bilinear models with double long memory

We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation X_t=ζ_tA_t+B_t,(), where are standard i.i.d. r.v.'s, and A_t,B_t are moving averages in X_s, s<t. Stationary solution of () is obtained as an orthogonal Volterra expansion. In the case A_t≡1, X_t is the classical AR(∞) process, while B_t≡0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, X_t may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters and , respectively. We also discuss the hyperbolic decay of auto- and/or cross-covariances of X_t and X_t² and the asymptotic distribution of the corresponding partial sums’ processes.     

Evaluating Currency Risk in Emerging Markets

We present a systematic approach to the problem of evaluating currency risk. The approach involves a test for stationarity, and a method of estimating Value-at-Risk (VaR) and Expected Shortfall (ES) from dependent heavy-tailed data. Various estimation methods are compared and the accuracy of the approach is discussed. An application of the technique to the Mexican peso/US dollar exchange rate reveals the level of currency risk foreign investors face in Mexico.      

A generalized fractionally differencing approach in long-memory modeling

We extend the class of fractional ARIMA models to the class of fractional ARUMA models, which describe long-memory time series with long-range periodical behavior at a finite number of spectrum frequencies. The exact asymptotics of the covariance function and the spectrum at the points of peaks and zeros are given. To obtain asymptotic expansions, Gegenbauer polynomials are used. Consistent parameter estimation is discussed using Whittle's estimate. Research supported by the Alexander von Humboldt Foundation. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 1, pp. 65–81, January–March, 1995.     

Development of an HPLC post-column antioxidant assay for Solidago canadensis radical scavengers

The aim of this work was to modify and validate the post-column high-performance liquid chromatography (HPLC)-ABTS and DPPH methods for evaluating the antioxidant activity of the methanolic extracts of Solidago canadensis (Canadian goldenrod) leaves and flowers. Separation of the analytes was performed via the HPLC-PDA method on a YMC analytical column using a gradient elution program. Three compounds with antioxidant properties – chlorogenic acid, rutin and isoquercitrin – and two unidentified antioxidants were established. The research showed that the coil temperature regimes and loop length combinations influence the optimised post-column assay method for detecting the antioxidant activity of goldenrod radical scavengers. Investigations established that the temperature in the reaction coil was a substantial factor contributing to the signal strength of the analytes after reacting with the DPPH and ABTS radicals.