(1/α)-Self similar α-stable processes with stationary increments

Originally published as a technical report no. 892, February 1990 for Cornell University Operations Research and Industrial Engineering. Available online: http://hdl.handle.net/1813/8775In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the α-stable Lévy motion is the only α-stable process with stationary increments if 0 < α < 1. We also introduce new classes of α-stable processes with stationary increments for 1 < α < 2.https://www.sciencedirect.com/science/article/pii/0047259X9090031C?via=ihubhttps://www.sciencedirect.com/science/article/pii/0047259X9090031C?via=ihubAccepted mansucrip

Heavy tails and electricity prices: Do time series models with non-Gaussian noise forecast better than their Gaussian counterparts?

This paper is a continuation of our earlier studies on short-term price forecasting of California electricity prices with time series models. Here we focus on whether models with heavy-tailed innovations perform better in terms of forecasting accuracy than their Gaussian counterparts. Consequently, we limit the range of analyzed models to autoregressive time series approaches that have been found to perform well for pre-crash California power market data. We expand them by allowing for heavy-tailed innovations in the form of α-stable or generalized hyperbolic noise.Electricity; price forecasting; heavy tails; time series; α-stable distribution; generalized hyperbolic distribution

A functional limit theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process

Under proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the M1-topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed symmetric α-stable Lévy process. The time change is given by the inverse β-stable subordinator

Monte Carlo-Based Tail Exponent Estimator

In this paper we study the finite sample behavior of the Hill estimator under α-stable distributions. Using large Monte Carlo simulations we show that the Hill estimator overestimates the true tail exponent and can hardly be used on samples with small length. Utilizing our results, we introduce a Monte Carlo-based method of estimation for the tail exponent. Our method is not sensitive to the choice of k and works well also on small samples. The new estimator gives unbiased results with symmetrical con_dence intervals. Finally, we demonstrate the power of our estimator on the main world stock market indices. On the two separate periods of 2002-2005 and 2006-2009 we estimate the tail exponent.Hill estimator, α-stable distributions, tail exponent estimation

Tail estimation of the stable index α

AbstractA refined tail-estimation procedure for measuring the index of stability of stable Paretian or α-stable distributions is proposed. The estimator is more suitable for α-stable laws than the widely used estimator proposed in [1]

Linear inference under matrix-stable errors

Linear inference is the foundation stone for much of theoretical and applied statistics. In practice errors often have excessive tails and are lacking the moments required in conventional usage. For random vector responses such errors often are modeled via spherical α-stable distributions with stability index α ϵ (0, 2], arising in turn through central limit theory but converging to non-Gaussian limits. Earlier work [Jensen, D.R. (2018). Biom. Biostat. Int. J. 7: 205–210] reexamined conventional linear models under n-dimensional α-stable responses, to the effect that Ordinary Least Square (OLS) solutions and residual vectors under α-stable errors also have α-stable distributions, whereas F ratios remain exact in level and power as for Gaussian errors. The present study generalizes those findings to include multivariate linear models having matrix responses of order (n×k). Topics in inference focus on both location and scale matrices, the latter in connection with analogs of simple, multiple, and canonical correlations without benefit of second moments, seen nonetheless to gauge degrees of association under α-stable symmetry

Sample path properties of stochastic processes represented as multiple stable integrals

Originally published as a technical report no. 871, October 1989 for Cornell University Operations Research and Industrial Engineering. Available online: http://hdl.handle.net/1813/8754This paper studies the sample path properties of stochastic processes represented by multiple symmetric α-stable integrals. It relates the “smoothness” of the sample paths to the “smoothness” of the (non-random) integrand. It also contains results about the behavior of the distribution of suprema and zero-one laws

Weak convergence of sums of moving averages in the α-stable domain of attraction

Skorohod has shown that the convergence of sums of i.i.d. random variables to an a-stable Levy motion, with 0 < a < 2, holds in the weak-J1 sense. J1 is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-J1 conver- gence cannot hold because adjacent jumps of the process can coalesce in the limit; however, if the moving average coefficients are positive, then the adjacent jumps are essentially monotone and one can have weak-M1 con- vergence. M1 is weaker than J1, but it is strong enough for the sup and inf functionals to be continuous

Stable fractal sums of pulses: the cylindrical case

A class of α-stable, 0\textlessα\textless2, processes is obtained as a sum of ’up-and-down’ pulses determined by an appropriate Poisson random measure. Processes are H-self-affine (also frequently called ’self-similar’) with H\textless1/α and have stationary increments. Their two-dimensional dependence structure resembles that of the fractional Brownian motion (for H\textless1/2), but their sample paths are highly irregular (nowhere bounded with probability 1). Generalizations using different shapes of pulses are also discussed

Continuous belief functions and α-stable distributions

International audienceThe theory of belief functions has been formalized in continuous domain for pattern recognition. Some applications use assumption of Gaussian models. However, this assumption is reductive. Indeed, some data are not symmetric and present property of heavy tails. It is possible to solve these problems by using a class of distributions called α-stable distributions. Consequently, we present in this paper a way to calculate pignistic probabilities with plausibility functions where the knowledge of the sources of information is represented by symmetric α-stable distributions. To validate our approach, we compare our results in special case of Gaussian distributions with existing methods. To illustrate our work, we generate arbitrary distributions which represents speed of planes and take decisions. A comparison with a Bayesian approach is made to show the interest of the theory of belief functions