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Two different approaches, called Type-I and Type-II, to linear least-squares prediction of a long-memory time series are distinguished. In the former, no new theory is required and a long-memory time series is treated on par with a standard shortmemory time series and its multistep predictions are obtained by using the existing modelling approaches to prediction of such time series. The latter, by...
Fractional Brownian motion is one of most cogent mathematical models for strongly correlated stochastic processes with self-similarity. In this article, we give a pedagogic introduction to this theory and investigate some of the statistical, geometric, and fractal properties of fractional Brownian motion and fractional Gaussian random fields. The connection between fractional Brownian motion and the...
This chapter offers, first, an introductory walk through the notions related to scaling phenomena and intuitions behind are gathered to formulate a tentative definition. Second, it introduces the mathematical model of self-similar processes with stationary increments, understood as the canonical reference to describe scaling. Then, it shows how and why the wavelet transform constitutes a powerful...
In this paper we review some recent results on the statistical properties of wavelet-based estimators of the Hurst parameter when the time series are heavy tailed with α-stable infinite variance distributions. We focus on two models: linear fractional stable motion (LFSM, in short) and fractional autoregressive moving average (FARIMA, in short) time series with SαS innovations. Using the wavelet coefficients...
The Lamperti transformation defines a one-to-one correspondence between stationary processes on the real line and self-similar processes on the real half-line. Although dating back to 1962, this fundamental result has further received little attention until a recent past, and it is the purpose of this chapter to survey the Lamperti transformation and its (effective and/or potential) applications,...
The partly random fractal sums of pulses (PFSP) are a family of random functions that depend on a kernel function K and at least two positive parameters C and δ. Given K, the construction of F(t; C, δ) consists in adding N affine versions of a pulse as follows. The pulse height Λ and its width W are random variables related by w/λδ = a constant. The width is distributed according to the...
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