It's not arsinus FUNCTION It's arcsinus PROBABILITY law (which contains an arcsinus function in it but it is not arcsinus function per se of course) from mathematician Paul Levy. As I warned it is recent law in statistical field so that you didn't learn it at school for some if not most of you and you won't find much about it on internet also ... except perhaps the biography of the guy http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Paul.html where it is said : "If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Lévy. " Paul Lévy is most well known for being the "inventor" also of "fat tails" distributions with no (fixed) mean nor (fixed) variance : it is since that also that people don't take Normal Law as the only "universal" or "natural" law any more (btw this a probability faq - work in progress only - where I remind why the term "natural" is used for Normal Law http://www.econometric-wave.com/faqs/probability/home.html.html)
counterintuitive yeah I will come back an other day to continue this thread - I am too busy at the moment, so perhaps not before the we.
How about these: http://scholar.lib.vt.edu/ejournals/JMSEC/v5n1/09344.pdf http://ftp.arl.mil/random/random.pdf
You're just saying that you're surprised that the guy in the forest gets way off (or stays off) course so often. I'm just saying that most people are not surprised by this result. They understand that, (1) given a long enough journey, they're likely to get blown off course sometime during it, and (2) once blown off course, they're likely to stay off course without help from some non-random source. They've apparently internalized the math you are presenting, based on the effects of being buffeted about by miscellaneous forces throughout their lives. ("staying off course" = staying on same side of projected point)
Harry, your massive written materials must be very good. Otherwise, how could it be possible that so many of us would spend time in reading (at least part of) them?
The density function of the arcsine distribution (as described in the link provided by OddTrader) depicts the results I obtained. This is a visual affirmation only.
Confusing persistency with "true trend" is like this author who confuses the edge with the realisation of this edge: http://gamblinglinks.com/art010228.html "If you're flat betting, i.e., betting the same amount on each even money bet: blackjack hand, roll of the dice, or spin of the wheel, you get an edge by winning more bets than you lose." If you launch a fair coin 10 times you can obtain for example 7 tails and 3 heads, but the edge of the coin flipping hasn't changed of course although you obtained more tails than heads for ONE realisation ! But you couldn't predict that BEFORE it occurs so that if you bet on tails, at the end of the ten times you will be a winner but it is still due to chance. And whatever strategy (martingale or "money management") you will use the edge of the system won't change, although there are better strategies than other but only for short term. I will talk about that in another thread (I will show how the "compensation law" which says that the difference between heads and tails should pass by 0 an INFINITE NUMBER OF TIMES as time elapses, used in martingale by gamblers, can be dangerous because of this "persistency law" - the two laws which appear to be antinomic are in fact related mathematically. This doesn't mean that the contrary of a martingale is better : it is also a martingale mathematically nevertheless some martingales can be better than others depending on some conditions).
The conclusion - of Berstein but it is not "his" conclusion it is statistical implication - is that judging performance of traders on equity curve is very difficult since equity curve is a cumulated sum that will deviate larger and larger from the zero line even if it is positive how can you judge by this sole criteria that it is not due to chance since chance can produce the same kind of results ? So an equity curve that is trending positively is a necessary condition (if it is not positive it isn't even worth of course ) but not sufficient condition. The problem is worsened if market doesn't follow a random walk because the jumps are rare events and that rare event are even more unpredictable stochastically. Random walk is the first kind of risk which is assimilated to usual noise, the jumps are a second kind of risk some calls uncertainty that is "more unpredictable" than the usual noise. That's what the so-called quants and fundamentalists base their reasoning to affirm that speculators who are winners are survival bias because as Berstein quoted Buffet in his book "if 215 millions of monkeys were playing the same game" the result would be the same - there will be some big winners and many losers of course but we don't just hear about the losers. It doesn't mean that these winners doesn't have any talent, for example by using money management (which is optimisation techniques and in fact just noble term for martingale term used by gamblers) but this money management cannot circumvent the edge for the global population. Of course you will find tremendous sucessful gamblers but for sure they are survival for nearly all of them since everybody knows that casinos are very carefull to give them no true edge. But the casinos know that they must let some players make big wins because it plays a role in their marketing so as to attract all the crowd. As for stock market, as long as it is not demonstrated academically that it is not so unpredictable, the market, for most traders, should be like the casinos, so that the big winners that have access to no special information are like the big winners in casinos: they would be survival bias - once again even if they have applied the optimal strategy it doesn't change the fact that the edge of the system was as it was: null or even negative so by pure logic any winner, in this hypothesis, has benefited from so called chance: chance is ONE realisation of a system that has no edge but that has deviated from the theorical value only due to sampling fluctuation. I just exposed above the quants and fundamentalists point of view which cannot be challenged easily on scientific basis by the others schools - once again even if market doesn't follow a normal random walk. I can give you one of the big reason : there could be no strict frontier between deterministic process and stochastic process ... so that trying to prove that something is not stochastic whereas such thing imitates stochastic process perfectly - by using traditional stochastic criterias like autocorrelations or others (like spectral analysis) for example one of known deterministic process called tent map has been proved to behave like a random process.