This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a suitable discretization of the Hausdorff theory of fractal dimension. We also find some connections between our definition and the classical ones and also with fractal dimensions I & II (see [7]). Therefore, we generalize them and obtain an easy method in order to calculate the fractal dimension of strict self-similar sets which are not required to verify the open set condition.

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In this paper, we explore the (in)efficiency of the continuum Bitcoin-USD market in the period ranging from mid 2010 to early 2019. To deal with, we dynamically analyse the evolution of the self-similarity exponent of Bitcoin-USD daily returns via accurate FD4 approach by a 512 day sliding window with overlapping data. Further, we define the memory indicator by the difference between the self-similarity exponent of Bitcoin-USD series and the self-similarity index of its shuffled series. We also carry out additional analyses via FD4 approach by sliding windows of sizes equal to 64, 128, 256, and 1024 days, and also via FD algorithm for values of q equal to 1 and 2 (and sliding windows equal to 512 days). Moreover, we explored the evolution of the self-similarity exponent of actual S&P500 series via FD4 algorithm by sliding windows of sizes equal to 256 and 512 days. In all the cases, the obtained results were found to be similar to our first analysis. We conclude that the self-similarity exponent of the BTC-USD (resp., S&P500) series stands above 0.5. However, this is not due to the presence of significant memory in the series but to its underlying distribution. In fact, it holds that the self-similarity exponent of BTC-USD (resp., S&P500) series is similar or lower than the self-similarity index of a random series with the same distribution. As such, several periods with significant antipersistent memory in BTC-USD (resp., S&P500) series are distinguished.

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