**4. Summary**

In this study, we analyzed high-frequency quotations of the CFD contracts associated with the stock market indices, the stocks themselves, and the selected commodities as well as with the most frequently traded currency exchange rates and the cryptocurrency prices. All of the data sets covered the years 2017–2020 except for the stock share CFDs, which covered the years 2018–2020. We analyzed the returns at a few different time scales from 1 s to 1 h and constructed the return distributions in order to investigate their tails. Our principal objective was to compare the tail behavior of the distributions derived from contemporary data with the behavior of the distribution tails in the past for the same assets. We applied the power-law function and the stretched exponential one to model the empirical distributions. A hypothesis that we planned to verify was the one formulated in [24,32,34], which states that, together with the acceleration of the information flow and processing across the financial markets, we can observe a significant change in the statistical properties of the returns at a particular time scale related to an effective acceleration of the market time with all of the possible consequences of this fact.

The results are mixed. On the one hand, the stock market indices (DJIA, DAX30, and S&P500, for which the present results can be compared directly with earlier works) do not show any further signatures of the time acceleration compared with the data from 1998–1999 and 2004–2006. It seems that the acceleration that was reported in [24,32] stopped or was only a temporary effect. Such effects were already reported before for Asian markets [35,69] as well as in this work regarding the stocks, so they may be a source of the observed behavior. On the other hand, the results for the individual stock groups

show that the market time acceleration can still be ongoing, but it is masked at the level of indices owing to the cross-correlations among the stocks that are now stronger and developing faster than even during the years 2004–2006 [32]. That particular time interval (2004–2006) was characterized by a volatility much smaller than in recent years, which witnessed large market events such as the flash crash on 5 February 2018, the coronavirusrelated unsteadiness in early 2020 and the subsequent rally ending with new record highs of S&P500 in August, the oil price drop in April 2020, etc. Large events, especially large falls, elevate the market correlation level, which can influence the statistical properties of data, including the distribution of returns. The auto- and cross-correlations are involved in an interesting interplay between two opposite-acting factors. The first factor is the market time flow speed, which works for market efficiency by shortening the period when the market autocorrelations are admissible. This factor shifts gradually the low-*α* behavior and the central limit theorem's realm to ever shorter time scales. The second factor is the asset cross-correlation strength, which causes thickening of the tails and decreases in *α* and *β*. It also violates the assumption of random variable independence and prevents the CLT from affecting the aggregated returns. This interplay and its consequences are interesting enough to be worthy of some more attention in future analyses. In particular, they can be responsible for the reported behavior of the return distributions in different time periods and suppressing the effects of the market time acceleration.

Currency exchange rates also no longer feel the market time acceleration such as that during 2004–2006 [32], but now, not only is there no further time scale shortening but also a moderate step backwards is observed: the inverse cubic scaling is seen at longer time scales than in 2004–2006 but is still significantly shorter than that during the years 1987–1993 [116]. The cryptocurrencies (BTC and ETH) show the same crossover scale as before—equal to 1 h [131]. Since this market is relatively young, it underwent a phase of strong market time acceleration after 2013, and now, it seems to be stabilized. It is still the market that shows the most exemplary inverse cubic scaling behavior across different scales out of all the markets analyzed in this work. Gold price CFDs show a clear difference between the present results and the distribution tails over the years 1969–1999 [37] and 2012–2018 [36] with increased tail slope during the recent years. In contrast, there is no clear change in the tail slope regarding silver, high-grade copper, and crude oil.

It should be noted, however, that the CFD contract price quotations analyzed here are not precisely the same as the related asset spot price quotations, which the authors of other works dealt with. This difference may partially account for the difference in the outcomes. Finally, the COVID-19 pandemic outburst that took place in March–April 2020 in the U.S. constituted a strong perturbation to all the markets, caused large-amplitude price fluctuations, and led to a strong increase in the cross-correlations among many assets. For example, it resulted in decreasing distribution tail slopes for the CFD returns for crude oil and gold. Even more significant were the bitcoin fluctuations, which become Lévy stable for the pandemic-outburst period.

In general, our results indicate that the monotonous shift in the time scales at which different types of dynamics can be observed in the financial data as well as the related continuously accelerating market time from past to present are oversimplified. In fact, there can be an underlying long-term trend of this type, but it is "decorated" with short-term phases of abrupt acceleration and, then, deceleration and stagnation. Our results indicate that the real market dynamics consists of continuous alternation of different regimes with different statistical properties that can form the overall impression of the market evolution direction. Together with the aforementioned problem of how the asset cross-correlations and the shortening autocorrelations compete against each other in shaping the statistical properties of data, it opens an intriguing direction for future work.

**Author Contributions:** Conceptualization, S.D., J.K. and M.W.; methodology, S.D., J.K. and M.W.; software, M.W.; validation, S.D., J.K. and M.W.; formal analysis, S.D., J.K. and M.W.; investigation, S.D., J.K. and M.W.; resources, M.W.; data curation, M.W.; writing—original draft preparation, J.K.; writing—review and editing, J.K. and M.W.; visualization, M.W.; supervision, S.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Publicly available datasets were analyzed in this study. This data can be found at [122,123].

**Conflicts of Interest:** The authors declare no conflict of interest.
