Extrapolation of stable random fields

Abstract

In this paper, we discuss three extrapolation methods for α-stable random fields with α ∈ (1, 2]. We justify them, giving proofs of the existence and uniqueness of the solutions for each method and providing sufficient conditions for path continuity. Two methods are based on minimizing the variability of the difference between the predictor and the theoretical value, whereas in the third approach we provide a new method that maximizes the covariation between these two quantities.

Conditional Moments of Noncausal Alpha-Stable Processes and the Prediction of Bubble Crash Odds

“…The new family of functions introduced contains functions related to the marginal density of the stable random variable Miller (1978) and Cambanis and Miller (1981) to replace the ill-defined covariance between two symmetric α-stable random variables, and has been a popular tool to formulate point forecasts of infinite variance α-stable processes [see Karcher et al (2013) and the references therein]. The new constants κp and λp, 2 p  introduced here, which intervene in the expressions of the higher order conditional moments of (X1, X2), can be seen as extending this dependence measure to higher…”

“…In special cases, expressions of the conditional expectation and variance have been obtained, and revealed that noncausal processes can feature GARCH type effects in calendar time despite such effects not being explicitly included in the modelling [Fries and Zakoian (2019)]. Provided the expressions of the conditional moments are derived, this suggests that point forecasts of noncausal processes based on their conditional expectation, variance, skewness and kurtosis could be formulated -as opposed to other predictors specifically introduced to circumvent the infinite variance of α-stable processes, such as minimum L  -dispersion or maximum covariation (see Karcher et al (2013) and the references therein).…”

Extrapolation of Stationary Random Fields

“…Proof. 2 For the proof of the existence and uniqueness of MCL we refer the reader to the paper [16]. It is also shown there that the vector of MCL weights…”

“…Figure 8(a) shows a realization of the stationary sub-Gaussian field with α = 0.8 and covariance function C of the Gaussian part as above. A Maximum Likelihood (ML) predictor for sub-Gaussian random fields is introduced in [16]. It is shown in Theorem 11 of that paper that LSL, COL and ML methods coincide if α ∈ (1, 2).…”

“…In Sections 2 and 3 we mainly follow the books [4,5,38,42]. Section 4 is based on the paper [16], it also contains some new results for stable fields with the infinite first moment, see Section 4.4.…”