**1. Introduction**

Studying the world cryptocurrency market is welcome for many reasons. Up to now, it constitutes the most spectacular and influential application of the distributed ledger technology called the blockchain, which, in the underlying peer-to-peer network, allows for the same access to information for all participants [1,2]. Research on blockchain technology is also unique because all related data are publicly available in the form of the history of every operation performed on the network. Furthermore, the tick-by-tick data for each transaction made on the cryptocurrency exchange are freely available using the application programming interfaces (APIs) of a given exchange.

As far as the financial, economic, and, in general terms, social aspects of cryptocurrencies are concerned, a basic related question that arises is whether such digital products can be considered as a commonly accepted means of exchange [3–5]. This is a complex issue involving many social, economical, and technological factors, such as trust, perceived risk, peer opinions, transaction security, network size effect, supply elasticity, and so on. However, also from a dynamical perspective, for this to apply, a certain level of maturity expressed in terms of market efficiency, liquidity, stability, size, and other characteristics is required [6,7]. Moreover, the developed markets show several statistical properties that newly established emerging markets often lack. Among such properties, one can list the so-called financial stylized facts: heavy tails of the probability distribution functions of fixed-time returns, long-term memory of volatility, a hierarchical structure of the asset cross-correlations, multifractality, and a stable (or meta-stable) price impact function [8–11].

**Citation:** Drozd˙ z, S.; Kwapie ´ ˙ n, J.; W ˛atorek, M. What Is Mature and What Is Still Emerging in the Cryptocurrency Market? *Entropy* **2023**, *25*, 772. https://doi.org/ 10.3390/e25050772

Academic Editor: Marcel Ausloos

Received: 21 April 2023 Revised: 4 May 2023 Accepted: 6 May 2023 Published: 9 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

There is growing quantitative evidence that the cryptocurrency market continuously advances on a route to maturity understood as sharing its statistical properties with the traditional financial markets. For instance, the most popular and oldest cryptocurrency, bitcoin (BTC), has passed through two stages of the shaping of its probability distribution function (pdf). It started as an extremely volatile asset with pdf tails that used to decline according to a power law, with the exponent reaching almost the Lévy-stable regime (the Lévy parameter *α* ≈ 2) on short time scales over the years 2012–2013, but then, already in the years 2014-2015, the tails of its pdf became thinner and reached the inverse cubic behavior that is observed universally in the traditional financial markets [12]. From that moment on, BTC has maintained this property over the subsequent years [6,13,14]. The difference between BTC and traditional assets is that the inverse cubic behavior of the BTC pdf tails was reported to be preserved up to much longer sampling intervals due to their less frequent trading [12]. Similar effects were seen for other major crypto currencies, such as ETH [12,15]. Since BTC and the other cryptocurrencies are traded on many independent platforms that differ in trading frequency, the pdf properties of the same cryptocurrency can be different on different platforms [6]. This is quite a unique trait of the cryptocurrencies not observed, for example, in the stock markets and Forex. Heavy pdf tails were also found in time series of volume traded in time units [16,17], even in the case of cryptocurrencies [18,19]. These two quantities—the log-returns and volume—are related to each other, because the size of a trade can have a profound impact on price variation: large trades lead to large price jumps on average (although this relation might be more subtle [20–22]). Some authors argue that price impact assumes a functional form with a square-root dependence of the log-returns on volume [23–25] but others are cautious [21,22,26].

The long-term memory of volatility fluctuations is responsible for the effect of volatility clustering, i.e., periods of a volatile market with large-amplitude fluctuations are interwoven with periods of relatively tranquil dynamics. In addition, the volatility autocorrelation is of a power-law form [27]. This property has been seen in all financial markets and has also been found in cryptocurrency dynamics [14]. The range of memory is comparable in this case with the range for the stock and Forex markets [28,29]. The scale-free form of the autocorrelation function is connected to fractality, which also requires long-term or long-range correlations to be self-similar. The log-return fluctuations for all the traditional financial markets studied so far show multiscaling together with some other quantities, such as inter-transaction times [30–32]. Consistently, multifractal properties have been observed in the cryptocurrency market returns and inter-transaction times for different assets [6,18,33–39]. Apart from univariate multiscaling, its bivariate version has also been reported between log-returns for different cryptocurrencies: BTC and ETH [40].

Apart from correlations in time, asset–asset cross-correlations play an important role in the shaping of the financial market structure as they lead to the emergence of the hierarchical organization of the markets as well as coupling between different markets [41–44]. While the hierarchical cross-correlations among the assets traded on the same market are a clear indicator of market maturity, the role of potential couplings between different markets must be interpreted with care. This is because either the independent dynamics of a market or the profound coupling of a market with the world's leading markets, being the two opposite cases, can potentially be interpreted in favor of market maturity. The former because independence can be viewed as strength and as a possibility for using the assets traded on such a market as a safe haven in hedging strategies [45,46], and the latter because it suggests that such a market is a well-rooted part of global financial markets. However, intuitively, neither of these extremes seems to represent the notion of maturity well enough. It is more justified to view market maturity as the ability to switch its dynamics between independence and compliance because such a behavior can better reflect the complexity that one may expect to be the property characterizing a developed market. This is why neither the effect of the cryptocurrency market decoupling from Forex reported in [29] nor the effects of the cryptocurrency market independence [47–51] and strong coupling between the cryptocurrencies and traditional financial markets reported in [52–55], respectively, can alone be a signature of maturity. It is

rather the opposite: only such flexible dynamics swinging between idiosyncrasy and a strong subjugation of the market to an actual global trend can be a manifestation of market maturity.

In this work, stress was put on the investigation of current statistical properties of cryptocurrency log-returns and volume from the perspective of how these properties differ from their counterparts in the traditional financial markets: the stock markets, Forex, and commodity markets. One has to be aware, however, that the statistical approach constitutes only a segment of the issues related to market maturity.
