All zeroes are not alike. Given an infinite number of possible outcomes, the chance of ALL possible outcomes is zero. Does that mean no outcome is possible? When you are dealing with infinity, you are not allowed to reject a possibility just because it has a zero chance of occuring. Infinity is different. This is the concept of "almost surely." See http://en.wikipedia.org/wiki/Almost_surely.
Mathematical Proof that Progressions will never Overcome a Negative Expectation Game
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What a great quote!
So a zero in the context of infinity is a different sort of zero?
I always thought that zero (as in void) was strangely related to infinity anyway. zg
ZERO - Wikipedia
So a zero in the context of infinity is a different sort of zero?
I always thought that zero (as in void) was strangely related to infinity anyway. zg
ZERO - Wikipedia
Related classic proof that 1=2. See: http://en.wikipedia.org/wiki/Mathematical_fallacy
One must be very careful when dealing with zero and infinity. You can 'prove' all kinds of things that aren't true.
One must be very careful when dealing with zero and infinity. You can 'prove' all kinds of things that aren't true.
Watching the video instead >>
The fascinating and tragic story of Georg Cantor, maybe the greatest mathematician of the 19th century, who came up with the Continuum Hypothesis that will unravel the mysteries of infinity. Only Cantor had a little secret - he thought god was speaking to him and saw himself as his messenger. -- VIDEO- 9min - Georg Cantor's Infinities (Continuum Hypothesis)
I see why a proof math of martingale doom even without limits is not immediately forthcoming. I always thought martingale was a cheesy novelty... now I am quite surprised by the amount of scientific interest taken in martingales, and that they continue to be taken 'seriously' in financial investing/trading, as well as a math tool in themselves. zg
See - The Splendors and Miseries of Martingales
Martingales played an important role in the study of randomness in the twentieth century. Jean Ville introduced martingales in the 1930s in order to improve Richard von Mises’concept of a collective, and Claus-Peter Schnorr made martingales algorithmic in the 1970s in order to advance the study of algorithmic randomness.
Martingales played an important role in the study of randomness in the twentieth century. Jean Ville introduced martingales in the 1930s in order to improve Richard von Mises’concept of a collective, and Claus-Peter Schnorr made martingales algorithmic in the 1970s in order to advance the study of algorithmic randomness.
From - http://www.jehps.net/juin2009/Crepel.pdf
It was at this point that Ville tried the notion of a martingale. He thought “martingale” might be an Italian name. The word was associated with a classical argument for all gambling systems being illusory. Time would have to be infinite. Governments engage in such illusions nowadays.
How about this question, then:
If a casino offered you (truly) unlimited credit, no table limits, and no time limit (you alone choose when to stop), on a game with known -EV (say, roulette), would you play it?
I sure as heck would! I think anyone would be a fool not to.
As for "almost surely", QFIT, would you then agree that Martingale would "almost surely" work, or at least has an infinitesimal chance of failure? Or, similarly precisely, that martingale will be successful with a probability of 1?
If a casino offered you (truly) unlimited credit, no table limits, and no time limit (you alone choose when to stop), on a game with known -EV (say, roulette), would you play it?
I sure as heck would! I think anyone would be a fool not to.
As for "almost surely", QFIT, would you then agree that Martingale would "almost surely" work, or at least has an infinitesimal chance of failure? Or, similarly precisely, that martingale will be successful with a probability of 1?
So how did all the PhDs and Quants forget and crash the global economy? zg
--> The Global Collapse was a MARTINGALE Play
--> The Global Collapse was a MARTINGALE Play
Why would anyone risk an infinite amount of money for one bet? If you lose, you would be condemned to sit at the table for all eternity. This is what's wrong with these ridiculous attempts at twisting reality into making progressions systems "work." You have to keep piling ridiculous assumption on top of ridiculous assumption until no one can recognize the original question.
I read the piece entitled, On the history of martingales in the study of randomness, as best a mathematical novice can. I noted this hopeful comment (quoted below) in the Epilogue on page 27 that makes me think that martingales will someday, perhaps already, take their place along side other predictive strategies in practical applications, including gambling. While I don't understand all its implications, it does seem to suggest that certain martingales can become a reliable and useful tool. It's all sort of enigmatic to me, such as the statement at the end of the selection below, "... the logic does not depend on there being enough bets to define probability distributions." What does this mean in the real world?
Most of the work on algorithmic randomness since the 1970s has been concerned with infinite sequences. But Kolmogorov was always more interested in finite random objects, because only finite objects can be relevant to our experience. Some of his ideas for using the theory of complexity in probability modeling were extended by his student Evgeny Asarin [1,2]. Martingales, which can have a finite or infinite horizon, have also recently been considered as a foundation for probabilistic reasoning independently of the classical axioms [71]. Instead of forbidding a nonnegative martingale to diverge to infinity in an infinite number of trials, one forbids it to multiply its initial capital by a large factor in a finite number of trials. Predictions are made and theorems proven by constructing martingales. Tests are conducted by checking whether martingales do multiply their initial capital handsomely. The picture that emerges is a little different from classical probability theory, because the logic does not depend on there being enough bets to define probability distributions. http://www.jehps.net/juin2009/BienvenuShaferShen.pdf
I was ...what's the word...flabbergasted... by the amount of attention that has been given by serious mathematicians to the concept of martingales in infinite and finite random series. Aslan, In awe of mathematicians