Quantitatively investigating the locally weak stationarity of modified multifractional Gaussian noise

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 ( H min , H max ) . It means that Corr [ r ( τ ; H ( t 1 ) ) , r ( τ ; H ( t 2 ) ) ] ≥ 0.70 if H ( t 1 ) , H ( t 2 ) ∈ ( H min , H max ) , where r ( τ ; H ( t 1 ) ) and r ( τ ; H ( t 2 ) ) are the autocorrelation functions of mmGn with H ( t 1 ) and H ( t 2 ) for t 1 ≠ t 2 , respectively, and Corr [ r ( τ ; H ( t 1 ) ) , r ( τ ; H ( t 2 ) ) ] is the correlation coefficient between r ( τ ; H ( t 1 ) ) and r ( τ ; H ( t 2 ) ) . 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.