# Signatures of the Crypto-Currency Market Decoupling from the Forex
## 3 Fundamental Notions of the Multifractal Formalism  
The methodology applied herein for addressing the multifractality aspects of time-series is based on the formalism of multifractal cross-correlation analysis (MFCCA) [35]. This method represents a consistent generalization of the detrended cross-correlation approach (DCCA) [36] together with its multifractal extension [37]. The MFCCA methodology, in brief, allows quantifying both the scaling properties of individual time-series and the degree of multifractal cross-correlation between pairs of any two time-series. This methodology, then, introduced in a natural and self-consistent manner the \(q\)-dependent detrended cross-correlation coefficient \(\rho_{q}\)[38], such that the same allows filtering out the degree of strength of cross-correlations when it varies with the size of fluctuations. Let us consider a pair of time-series \(x(i)_{i=1,\ldots,T}\) and \(y(i)_{i=1,\ldots,T}\) divided into \(2M_{s}\) disjoint boxes having length \(s\) (i.e., \(M_{s}\) boxes starting from the opposite ends). The detrending procedure is then applied by calculating the residual signals \(X,Y\) within each box \(\
u\) (\(\
u=0,...,2M_{s}-1\)). These are equal to the difference between the integrated signals and the \(m\)th-order polynomials \(P^{(m)}\) fitted to the same, namely  
\[X_{\
u}(s,i)=\sum_{j=1}^{i}x(\
u s+j)-P^{(m)}_{X,s,\
u}(j), \tag{1}\]  
\[Y_{\
u}(s,i)=\sum_{j=1}^{i}y(\
u s+j)-P^{(m)}_{Y,s,\
u}(j). \tag{2}\]  
In typical cases, an optimal choice is provided by \(m=2\)[39; 40], which is retained in the present work. Figure 4: (Color online) Log-log plot of the cumulative distributions of the normalized absolute returns \(|r_{\Delta t}(t)|\) for BTC/EUR, BTC/USD, BTC/ETH, ETH/EUR, ETH/USD over the period between 1 July 2016 and 31 December 2018. The dashed line represents the expected inverse cubic power-law. Next, the covariance and the variance of \(X\) and \(Y\) within a box \(\
u\) are calculated according to  
\[f^{2}_{XY}(s,\
u) =\frac{1}{s}\sum_{i=1}^{s}X_{\
u}(s,i)Y_{\
u}(s,i), \tag{3}\] \[f^{2}_{ZZ}(s,\
u) =\frac{1}{s}\sum_{i=1}^{s}Z_{\
u}^{2}(s,i), \tag{4}\]  
where \(Z\) corresponds to either \(X\) or \(Y\). These quantities are subsequently used in defining a family of fluctuation functions having order \(q\)[35]:  
\[F^{q}_{XY}(s) =\frac{1}{2M_{s}}\sum_{\
u=0}^{2M_{s}-1}\text{sign}\left[f^{2}_{ XY}(s,\
u)\right]|f^{2}_{XY}(s,\
u)|^{q/2}, \tag{5}\] \[F^{q}_{ZZ}(s) =\frac{1}{2M_{s}}\sum_{\
u=0}^{2M_{s}-1}\left[f^{2}_{ZZ}(s,\
u) \right]^{q/2}. \tag{6}\]  
The real-valued parameter \(q\) plays the role of a filter, in that it amplifies or suppresses the intra-box variances and covariances such that, for large positive \(q\)-values, only the boxes (of size \(s\)) with the highest fluctuations contribute substantially to the sums in Equations (5) and (6), whereas, for negative \(q\)-values, only the boxes containing the smallest fluctuations provide a dominant contribution. The power-law dependence of \(F^{q}_{XY}(s)\) on \(s\) through the relation  
\[F^{q}_{XY}(s)^{1/q}=F_{XY}(q,s)\sim s^{\lambda(q)}, \tag{7}\]  
is considered as a manifestation of the fractal character of the cross-correlations. Here, \(\lambda(q)\) is an exponent that quantitatively characterizes the various fractality aspects. The \(q\)-independence of \(\lambda(q)\), then, reflects monofractality of the cross-correlations. Contrariwise, the \(q\)-dependence of \(\lambda(q)\) signals their more complex, multifractal character. For a single time-series, possible scaling of the corresponding \(F_{ZZ}\) reflects its scale-free properties  
\[F^{q}_{ZZ}(s)^{1/q}=F_{ZZ}(q,s)\sim s^{h(q)}, \tag{8}\]  
where \(h(q)\) stands for the generalized Hurst exponent. For multifractal signals, \(h(q)\) is a decreasing function of \(q\).