Relative cluster entropy for power-law correlated sequences

We propose an information-theoretical measure, the \textit{relative cluster entropy} $\mathcal{D_{C}}[P \| Q] $, to discriminate among partitions characterised by probability distribution functions $P$ and $Q$. The measure is illustrated with clusters generated pairs of fractional Brownian motions Hurst exponents $H_1$ $H_2$ respectively. For subdiffusive, normal superdiffusive sequences, relative entropy sensibly depends on difference between $H_2$. By using \textit{minimum principle, sequences characterized different correlation degrees are distinguished optimal exponent selected. As a case study, real-world market price series compared those obtained from fully uncorrelated (simple Browniam motions) assumed as model. yields $H_1=0.55$, $H_1=0.57$, $H_1=0.63$ respectively for prices DJIA, S\&P500, NASDAQ: clear indication non-markovianity. Finally, we derive analytical expression outcomes discussed arbitrary power-laws continuous random variables.