Power-law correlations, related models for long-range dependence and their simulation

Abstract

Martin and Walker ((1997) J. Appl. Prob. 34, 657-670) proposed the power-law ρ(v) = c|v|^-β, |v| ≥ 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 ρ(v) = (1 + |v/c|^α)^-β/α 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.