*2.2. Cumulative Distribution Functions of Returns and Volume*

The cumulative distribution function (cdf) *P*(*X* > *r*Δ*t*) can be calculated from the normalized returns *r*Δ*t*(*ti*)=(*R*Δ*t*(*i*) − *μ*)/*σ*, with *μ* and *σ* denoting sample mean and standard deviation, respectively. A form of this distribution varies among the markets and assets, but some interesting properties can be observed. There are generally three factors that shape it: the first one is liquidity, the second one is trading speed, and the third one is the overall market volatility [59]. If one focuses on a specific market, the most liquid assets show a faster decline in *P*(*X* > *r*Δ*t*) with *r*Δ*<sup>t</sup>* than the less liquid ones for a given Δ*t* [60]. However, most of the assets traded on mature markets reveal a power-law dependence of *P*(*X* > *r*Δ*t*) for some range of Δ*t* [23,27,60–62]:

$$P(X > r\_{\Delta t}) \sim |r\_{\Delta t}|^{ - \gamma},\tag{1}$$

with *γ* ≈ 3. It is observed for short sampling intervals and it is persistent for a range of Δ*t* due to the existence of strong inter-asset correlations. This inverse cubic power-law dependence breaks for sufficiently long Δ*t* and the cdf tails converge to the expected normal distribution. The speed of information processing on a given market also has influence on the crossover Δ*t*. Since this speed increases with time as new technologies enter the service, we observe a gradual decrease in the crossover Δ*t* across decades. The speed of market trading allows for a larger transaction number in time units, so this factor accelerates the market time even more [60]. The emerging markets, where investment strategies require the accommodation of significant risk, are thus highly volatile. The cdfs of the asset returns in this case often show heavy tails with the scaling exponent *γ* 3, sometimes even in the Lévy-stable regime. In such markets, the inverse cubic behavior of *P*(*X* > *r*Δ*t*) may occur for some assets only, whereas, for the other assets, it cannot be found at all. This is why such extreme tails are often considered to be an indicator of market immaturity [14].

Based on the average inter-transaction time *δt*, we categorized the considered cryptocurrencies into three groups: I, the most frequently traded cryptocurrencies (*δt* < 1*s*); II, the cryptocurrencies with the average trading frequency (1*s* ≤ *δt* < 2*s*); and III, the least frequently traded cryptocurrencies (*δt* ≥ 2*s*). Then, we calculated the average cdfs for the cryptocurrencies belonging to each group. We show these cdfs in Figure 2 (left panel, dotted lines) together with the cdfs for a few selected individual cryptocurrencies (solid lines). Their form can be compared with the inverse cubic power-law model denoted by a dashed line. It can be seen that the average distributions have their tail close to a power law, with the exponent *γ* being close to 3. The most liquid cryptocurrencies—BTC and ETH—develop tails that show a cross-over from the power-law regime to a CLT-like regime for relatively small values of |*r*Δ*t*| compared to both the average cdfs and to less frequently traded individual cryptocurrencies such as FUN, PERL, and WAN. The case of Dogecoin, which has the smallest slope in the middle of the distribution and, at the same time, does not have the thickest tail, is special. On the one hand, it can be included among the main cryptocurrencies due to the high frequency of transactions and capitalization, and, on the other, it was the subject of possible price manipulation through Elon Musk's tweets [63,64].

**Figure 2.** Cumulative distribution functions of the absolute normalized log-returns *r*Δ*<sup>t</sup>* (**left**) and the normalized volume traded *v*Δ*<sup>t</sup>* (**right**) for Δ*t* = 1 min in units of the respective standard deviations *σ* for the selected cryptocurrencies with the highest liquidity (BTC and ETH) or the heaviest tails (DOGE, FUN, PERL, and WAN). The average cumulative distribution functions for the cryptocurrencies with the average inter-transaction time fulfilling the relations *δt* < 1*s* (Group I, dotted red), 1*s* ≤ *δt* < 2*s* (Group II, dotted blue), and *δt* ≥ 2*s* (Group III, dotted green) are also shown. Power laws with the scaling exponents *γ* and *β* assuming values typical for the financial markets—*γ* = 3 and *β* = 3/2—are denoted by dashed lines. There is also a stretched exponential function fitted to the *v*Δ*<sup>t</sup>* distributions for BTC and ETH on the right (black dotted line).

Another quantity that is frequently observed to be power-law-distributed is normalized volume traded in time unit *v*Δ*t*(*ti*)=(*V*Δ*t*(*i*) − *μ*)/*σ* [16,23]:

$$P(X > v\_{\Delta t}) \sim v\_{\Delta t}^{-\beta}.\tag{2}$$

In this case, the exponent is much lower than for the absolute returns and corresponds to the Lévy-stable regime: *β* < 2. It was argued that there exists a simple relation between both the exponents: *β* = *γ*/2 [23]. Figure 2 (right panel) shows the cumulative distribution functions for *v*Δ*<sup>t</sup>* for the same individual cryptocurrencies and their Groups I-III as in Figure 2 (left panel). Now, the cdfs for BTC and ETH do not develop power-law tails. A model that best fits them is the stretched exponential function *<sup>P</sup>*(*<sup>X</sup>* <sup>&</sup>gt; *<sup>v</sup>*Δ*t*) <sup>∼</sup> exp *<sup>σ</sup>*−*<sup>η</sup> v* with *η* = 0.43. However, in the case of less frequently traded cryptocurrencies, which belong to Group III, one can observe the power-law relation. What makes the results obtained here different from their counterparts for, for instance, the stock markets, is that one does not find any cryptocurrency with its cdf being a power law with the exponent 3/2; the cdf tails decrease considerably faster here.
