Actually there are many different approaches to prove randomness (academic) or disprove randomness (fund managers to persuade their clients or their bosses ;-) in financial markets.
One approach I find especially interesting is based on algorithmic information theory. Basically what that does is to find an algorithm to compress financial data. The fewer regularities (= randomness) the more complex the algorithm will be. While e.g. $01010101$ will just be "repeat $01$ four times", $11010010$ seems to be "more random" so that the resulting algorithm will be more complex.
Brandouy, Olivier and Delahaye, J. P. and Ma, L., Estimating the Algorithmic Complexity of Stock Markets (May 1, 2011). International Conference of the French Finance Association (AFFI), May 11-13, 2011; Algorithmic Finance 2015, 4:3-4, 159-178. Available at SSRN: https://ssrn.com/abstract=1836886 or http://dx.doi.org/10.2139/ssrn.1836886
Randomness and regularities in finance are usually treated in probabilistic terms. In this paper, we develop a different approach in using a non-probabilistic framework based on the algorithmic information theory initially developed by Kolmogorov (1965). We develop a generic method to estimate the Kolmogorov complexity of numeric series. This approach is based on an iterative “regularity erasing procedure” (REP) implemented to use lossless compression algorithms on financial data. The REP is found to be necessary to detect hidden structures, as one should “wash out” well-established financial patterns (i.e. stylized facts) to prevent algorithmic tools from concentrating on these non-profitable patterns. The main contribution of this article is methodological: we show that some structural regularities, invisible with classical statistical tests, can be detected by this algorithmic method. Our final illustration on the daily Dow-Jones Index reveals a weak compression rate, once well- known regularities are removed from the raw data. This result could be associated to a high efficiency level of the New York Stock Exchange, although more effective algorithmic tools could improve this compression rate on detecting new structures in the future.
Markets seem to be quite efficient!