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Mathematical theory The mathematical characterisation of stock market movements has been a subject of intense interest. The conventional assumption has been that stock markets behave according to a random log-normal distribution.<ref name="ARWDWS">{{#invoke:citation/CS1|citation |CitationClass=book }}</ref> Among others, mathematician Benoît Mandelbrot suggested as early as 1963 that the statistics prove this assumption incorrect.<ref>The (Mis-)Behavior Of Markets</ref> Mandelbrot observed that large movements in prices (i.e. crashes) are much more common than would be predicted in a log-normal distribution. Mandelbrot and others suggest that the nature of market moves is generally much better explained using non-linear analysis and concepts of chaos theory.<ref>'Father of Fractals' takes on the stock market</ref> This has been expressed in non-mathematical terms by George Soros in his discussions of what he calls reflexivity of markets and their non-linear movement.<ref>Soros, G. Alchemy of Finance, Wiley Investment Classics. 2003</ref> George Soros said in late October 1987, 'Mr. Robert Prechter's reversal proved to be the crack that started the avalanche'.<ref>Marketwatch.com</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=news }}</ref>

Research at the Massachusetts Institute of Technology suggests that there is evidence the frequency of stock market crashes follows an inverse cubic power law.<ref>Stock trade patterns could predict financial earthquakes</ref> This and other studies such as Prof. Didier Sornette's work suggest that stock market crashes are a sign of self-organized criticality in financial markets.<ref>Didier Sornette, Professor of Geophysics</ref> In 1963, Mandelbrot proposed that instead of following a strict random walk, stock price variations executed a Lévy flight.<ref>The variation of certain speculative prices</ref> A Lévy flight is a random walk that is occasionally disrupted by large movements. In 1995, Rosario Mantegna and Gene Stanley analyzed a million records of the S&P 500 market index, calculating the returns over a five-year period.<ref>Scaling behaviour in the dynamics of an economic precursors and replicas." Journal de Physique I France 6, No.1, pp. 167–175.</ref> Researchers continue to study this theory, particularly using computer simulation of crowd behaviour, and the applicability of models to reproduce crash-like phenomena.

Research at the New England Complex Systems Institute has found warning signs of crashes using new statistical analysis tools of complexity theory. This work suggests that the panics that lead to crashes come from increased mimicry in the market. A dramatic increase in market mimicry occurred during the whole year before each market crash of the past 25 years, including the recent financial crisis. When investors closely follow each other's cues, it is easier for panic to take hold and affect the market. This work is a mathematical demonstration of a significant advance warning sign of impending market crashes.<ref>Predicting economic market crises using measures of collective panic arXiv:1102.2620v1 [q-fin.ST]</ref><ref>Possible Early Warning Sign for Market Crashes</ref>

The Hindenburg Omen, developed by physics professor Jim Miekka, is a controversial indicator that is believed by many to predict stock market crashes.

A recent phenomenon, known as the RR Reversal, has also been well documented in recent years - where a rapidly increasing stock experiences an inexplicable and sudden pullback to the magnitude of 10 - 40% within a month. {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}


Stock market crash sections
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Mathematical theory
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