The demonstration has been done by Levy: the probability P(Kn/n < a) is the sum of combinatorial terms where Kn is the number of times when the player's gain is positive and it tends towards 1/Pi * Arcsinus , the density is just derivation and it is because this density is infinite when x = 0 or 1 (since the denominator is x(x-1)) that 1/2 of the time is not the most probable but rather 0 (none of the time) or 1 (all of the time).
Now there is something related to this arsinus law that the quants and fondamentalists didn't take into account when they compare Stock market to casinos but hold on I'm tired for now I will come back another day .
Your site (from my last visit) looks great! Perhaps more down-to-earth applications would be even better!
Good points Harry. I was just thinking, if as they say, only 10% of traders are successful, one has to wonder what percentage of that top 10% have been successful by pure chance. The real % of "good" traders could be much lower than 10%. :eek:
It's rather that at long term there are only a few percentage of "good traders" but on short term (this is relative of course to period and style of trading this means dozens of years for portfolio managers for example to a few months/years for short term traders) there can be even a majority of winners ... and after that a majority of loosers: that's what persistency means, it's easy to see it during bull phase where all the public and traders believe into a "new era" because it seems to last long time enough (remember arcsinus law is about time not about the distribution of mean which is another law - the central limit theorem) and that everybody seems to be winner . Since at each cycle there are newcomers the same illusion can begin again and again, this on multiple scales.
What if Irving Fisher the most famous economist of his time before 1929 had known the persistency law (Levy discovered and demonstrated the law in 1939 only) ? Would he had declared that "stock prices were not overinflated but, rather, had achieved a new, permanent plateau." http://cepa.newschool.edu/het/profiles/fisher.htm "This Yale economist was an eccentric and colorful figure. When Irving Fisher wrote his 1892 dissertation, he constructed a remarkable machine equipped with pumps, wheels, levers and pipes in order to illustrate his price theory - see here for pictures of his draft and his first and second prototypes. Socially, he was an avid advocate of eugenics and health food diets. He made a fortune with his visible index card system - known today as the rolodex - and advocated the establishment of an 100% reserve requirement banking system His fortune was lost and his reputation was severely marred by the 1929 Wall Street Crash, when just days before the crash, he was reassuring investors that stock prices were not overinflated but, rather, had achieved a new, permanent plateau."
Kenneth Galbraith like to refer to Fisher's prediction as his "immortal estimate" . It's all the more funny that Fisher has created with 2 others the so-called Cowles Commission (see its history http://cowles.econ.yale.edu/reports/20yr/his_1.htm) which is at the very origin of the RMH (Random Market Hypothesis). In 1933 they published an article untitled "Can Stock Market Forecasters forecast ?" their conclusion is that it is doubtful. Notably the Cowles commission has attacked Hamilton father of so-called Dow Theory - Hamilton who has formalised Dow's idea - saying that his forecast results was no more or less good than if it was due to chance .