Distinguishing Stationary/Nonstationary Scaling Processes Using Wavelet Tsallis 𝑞-Entropies

Table 1

Wavelet spectrum or wavelet variance associated with different types of scaling processes. 𝔼(⋅), Var(⋅), and Ψ(⋅) represent expectation, variance, and Fourier integral operators, respectively.

Type of scaling process	Associated wavelet spectrum or variance
Long-memory process of index 𝛼	𝔼𝑑^2𝑋(𝑗,𝑘)∼2^𝑗𝛼𝐶(𝜓,𝛼), 𝐶(𝜓,𝛼)=𝑐𝛾∫|𝑓|^−𝛼|Ψ(𝑓)|^2𝑑𝑓
Self-similar process of index 𝐻	𝔼𝑑^2𝑋(𝑗,𝑘)=2^𝑗(2𝐻+1)𝔼𝑑^2𝑋(0,𝑘)
Hsssi process of index 𝐻	Var𝑑^2𝑋(𝑗,𝑘)=2^𝑗(2𝐻+1)Var𝑑𝑋(0,0)
Pure power-law process of index 𝛼	𝔼𝑑^2𝑋(𝑗,𝑘)=𝐶2^𝑗𝛼