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Power-law correlations, related models for long-range dependence and their simulation. (English) Zbl 0972.62079
Summary: R.J. Martin and A.M. Walker [J. Appl. Probab. 34, No. 3, 657-670 (1997; Zbl 0882.62085)] proposed the power-law \(\rho(v)= c|v|^{-\beta}\), \(|v|\geq 1\), as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of \(c\) is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family \(\rho(v)= (1+|v/c|^\alpha)^{-\beta/ \alpha}\) allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G10 Stationary stochastic processes
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