*2.5. Multiscaling of Returns*

If the bivariate or univariate fluctuation functions can be approximated by a power-law relation

$$F\_q^{AB}(\mathbf{s}) \sim s^{h(q)},\tag{10}$$

where *h*(*q*) is a non-increasing function of *q* called the generalized Hurst exponent [77] and *A* and *B* stand for either *R* or *V*, the time series under study reveal a fractal structure. If *h*(*q*) = const = *H*, it means that this structure is monofractal, with *H* equal to the Hurst exponent, which is a measure of persistence; otherwise, it is multifractal [77]. Multifractal signals are governed by processes with long-range autocorrelations, which is why both properties are often observed together [78–81]. It is the case, for example, in financial data. If the relation (10) exists, it can be seen in double-logarithmic plots of *FAB <sup>q</sup>* (*s*). Figure 8 displays *FRR <sup>q</sup>* (*s*) for six cryptocurrencies, with −4 ≤ *q* ≤ 4 and 10 ≤ *s* ≤ 25,000. Out of these, four cryptocurrencies show unquestionable power-law functions—BTC, ETH, DOGE, and FUN—for all used values of *q* and for at least a decade-long range of scales, whereas PERL and WAN do not. The same result can be expressed in a different way by calculating the singularity spectra *f*(*α*) from *h*(*q*) according to the following relations:

$$\alpha = h(q) + q \frac{dh(q)}{dq}, \quad f(\alpha = q(\alpha - h(q)) + 1. \tag{11}$$

The Hölder exponents *α* quantify the singularity strength and *f*(*α*0) expresses the fractal dimension of a subset of singularities with strength *α* = *α*0. While many theoretical singularity spectra are symmetric, in a practical situation, one observes spectra that are asymmetric [14,28,31,82–85]. The insets in Figure 8 show *f*(*α*) calculated from *FRR <sup>q</sup>* (*s*) in the scaling regions of *s*. All the presented spectra are left-side asymmetric (their left shoulder, corresponding to positive *q*s, is longer). The origin of such a behavior can be twofold: the signals can develop heavy tails of the probability distribution functions that are unstable in the sense of Lévy yet their convergence to the normal distribution is slow [76], and the signals can be mixtures of processes that have different fractal properties: large fluctuations can be associated with a multifractal process (e.g., a multiplicative cascade), whereas small fluctuations can be monofractal. It often happens that the small fluctuations in financial time series are noise whereas the medium and large fluctuations carry meaningful information.

**Figure 8.** (Main plots) Univariate fluctuation functions *FRR <sup>q</sup>* (*s*) calculated from the log-returns *R*Δ*t*(*t*) with Δ*t* = 1 min for BTC, ETH, DOGE, FUN, PERL, and WAN. The breakdown of scaling for small scales and negative values of *q* in some plots is an artifact related to long sequences of zero returns in time series. (Insets) Singularity spectra *f*(*α*) calculated from the corresponding fluctuation functions in the range denoted by dashed red lines (if possible).

It was reported in the literature that cryptocurrencies can also show such asymmetric *f*(*α*) spectra [6,14]. From the perspective of this study, it is interesting to note that the spectra for BTC calculated for different historical periods show an elongation of the right shoulder of *f*(*α*) that corresponds to small fluctuations. It can be interpreted as a gradual building of a multifractal structure in BTC price fluctuations that started from large returns only in the early stages of BTC trading and were imposed on the smaller returns as the cryptocurrency market goes toward maturity. If one looks at Figure 8, BTC, ETH, and, to a lesser degree, DOGE—that is, the cryptocurrencies that are among the most capitalized ones—have noticeable right wings of *f*(*α*), whereas the more exotic cryptocurrencies, such as FUN, PERL, and WAN, do not develop the right wing at all. In agreement with what has been said before, despite various cryptoassets being traded on the same platforms, different ones can be found at different stages of the maturing process due to the different trading frequencies. This difference can also be observed in the possible scaling range of the fluctuation functions in Figure 8. In the case of the two most liquid cryptocurrencies, BTC and ETH, the *FRR <sup>q</sup>* (*s*) scaling can be observed almost from the beginning of the scale range, whereas, in the case of less liquid cryptocurrencies, the range of satisfactory scaling is significantly shorter and *FRR <sup>q</sup>* (*s*) even becomes singular on short scales due to the number of consecutive 1 min bins with zero returns. This is typical behavior in the case of less liquid financial instruments [14].
