## 3 Results  
We begin the presentation by taking a look at the time evolution of the quantities of interest over the interval considered in this work. Figure 1 (top) shows the price course of BTC and ETH expressed in tether (USDT)--a stablecoin pegged (1:1 on average) to the US dollar [43]. This stablecoin is used by trading platforms as a proxy for fiat currencies, because it facilitates trading and allows the market participants to avoiding taxation while temporarily closing the cryptocurrency positions. The lowest prices of BTC and ETH during the observed 26 months were 6202 USDT and 130 USDT, respectively, recorded on 1 April 2020, and the highest prices were 68,789 USDT, recorded for BTC on 10 November 2021 and 4892 USDT for ETH, which was recorded 6 days later. The time span considered comprised almost the entire rally and subsequent fall of this market. Figure 1: Evolution of the quantities of interest over the time period considered in this study for two principal cryptocurrencies: BTC (red circles) and ETH (blue squares). (**a**) Price \(P(t)\) of the cryptocurrencies expressed in US dollars; (**b**) logarithmic returns \(r_{\Delta t}(t)\) for \(\Delta t=10\) s; (**c**) mean volume traded \(\langle V_{\Delta t}(t)\rangle\) in 10 s intervals; (**d**) mean number of transactions \(\langle N_{\Delta t}(t)\rangle\) in \(\Delta t=10\) s. The averaging was carried out over a rolling window of 1 month with a step of 6 days. The logarithmic returns presented in Figure 1 (upper middle) are defined as  
\[r_{\Delta t}(t_{i})=\ln P(t_{i+1})-\ln P(t_{i}), \tag{6}\]  
where \(P(t_{i})\) is an asset price at time \(t_{i}\). Inferring their heavy-tailed probability distribution function and volatility clustering is straightforward. Periods of market turbulence are associated with a high amplitude of returns. In Figure 1, a rolling window of 1 week is applied to calculate the mean volume traded \(\langle V_{\Delta t}(t)\rangle\) (lower middle), and the mean number of transactions executed in 10 s-long intervals (bottom). The market sensed the most increased volatility level in September 2020, January-February 2021, and May-June 2021. These volatile periods can be associated with a few bear phases of the cryptocurrency market. The periods of increased volatility overlap with the periods of increased volume and increased number of transactions, as can be seen in Figure 1.  
These associations can be quantified in terms of the Pearson cross-correlation coefficient  
\[C_{\text{XY}}=\frac{(1/T)\sum_{i=1}^{T}(X_{i}-\langle X\rangle)(Y_{i}-\langle Y \rangle)}{\sigma_{\chi}\sigma_{\Upsilon}}, \tag{7}\]  
where \(\langle\cdot\rangle\) is mean and \(\sigma\) is standard deviation of the time series \(\{X_{i}\}\) and \(\{Y_{i}\}\). Values of \(C_{\text{XY}}\) for all pairs of the non-detrended time series are collected in Figure 2, showing that the strongest cross-correlations are observed for the absolute values of logarithmic returns of BTC and ETH (\(C_{\text{XY}}=0.72\)) and for the mean number of transactions and the mean volume traded of BTC (\(C_{\text{XY}}=0.75\)). All the related time series representing BTC are substantially cross-correlated among themselves, and the same can be noted regarding the time series representing ETH. The least cross-correlated are the pairs in which time series represent both cryptocurrencies, but even in this case, the obtained values are statistically significant. Now, we pass on to an analysis of the detrended time series. First, let us examine the fractal autocorrelations in terms of univariate fluctuation functions \(F_{q}^{\text{XX}}(s)\). The results for all time series of interest are plotted in Figure 3. All plots exhibit clear scaling for 2-3 decades for a range of \(q\), which indicates that the time series are fractal. Moreover, a 'broom'-like shape of the function plots for different values of \(q\) may be interpreted as a signature of the multifractal nature of time series.