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Quantitatively investigating the locally weak stationarity of modified multifractional Gaussian noise

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  • Li, Ming
  • Zhao, Wei

Abstract

We suggest that there exists a critical point H=0.70 of the local Hölder exponent H(t) for describing the weak stationary (stationary for short) property of the modified multifractional Gaussian noise (mmGn) from the point of view of engineering. More precisely, when H(t)>0.70 for t∈[0,∞], the stationarity of mmGn is conditional, relying on the variation ranges of H(t). When H(t)≤0.70, on the other side, mmGn is unconditionally stationary, yielding a consequence that short-memory mmGn is stationary. In addition, for H(t)>0.70, we introduce the concept of stationary range denoted by (Hmin,Hmax). It means that Corr[r(τ;H(t1)),r(τ;H(t2))]≥0.70 if H(t1), H(t2)∈(Hmin,Hmax), where r(τ;H(t1)) and r(τ;H(t2)) are the autocorrelation functions of mmGn with H(t1) and H(t2) for t1≠t2, respectively, and Corr[r(τ;H(t1)),r(τ;H(t2))] is the correlation coefficient between r(τ;H(t1)) and r(τ;H(t2)). We present a set of stationary ranges, which may be used for a quantitative description of the local stationarity of mmGn. A case study is demonstrated for applying the present method to testing the stationarity of a real-traffic trace.

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  • Li, Ming & Zhao, Wei, 2012. "Quantitatively investigating the locally weak stationarity of modified multifractional Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6268-6278.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:24:p:6268-6278
    DOI: 10.1016/j.physa.2012.07.043
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    References listed on IDEAS

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    1. Myoungji Lee & Marc G. Genton & Mikyoung Jun, 2016. "Testing Self-Similarity Through Lamperti Transformations," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 426-447, September.
    2. Setty, V.A. & Sharma, A.S., 2015. "Characterizing Detrended Fluctuation Analysis of multifractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 698-706.
    3. Li, Ming, 2020. "Multi-fractional generalized Cauchy process and its application to teletraffic," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).

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