The reason why the "best predictor" of the sign of x(i+1) - x(i) must be opposite to x(i) - 0.5 is due to anti-correlation and that x(i) is a UDRV in [0,1]. Equation (3) is wrong, anti-correlation is not -1/4 in that example. Clearly the authors screwed up the calculation of E[x(i)^2] - 3rd term in expansion. I can assure you it is not 1/2 for a UDRV in [0,1]. But the derivation for general distributions looked correct last time I went through this.
Yeah... that's a good starting point. What you're likely to find is that this approach will be often buying at bid and selling at ask; i.e. you'll "profit" from the spread which is, as we all know, a bit more difficult than simple stats No way on elimination of fat tails though... that's where some clever use of certain distributions will come in handy. Those tails are where all the opportunity is at, never ever try to get rid of them. Haha, yes this game is an interesting permutation of Monty Hall.
Optimal Stopping + Monty Hall is a good way of talking about it. The motivation is primarily the application of non-intuitive stats. If one were to try to convince someone of the theory behind monty hall they might have a difficult time... understanding random variables is not an easy task right off the bat. I remember a former roommate back in college was applying for a job and they asked him to write a computer program that was similar to the monty hall problem. This was back when I had never heard of anything about stats/monty hall, and, for a while, I thought the question was a trick question (mind I was 18 and had never taken a stats class). After seeing it work, it really grabbed my interest. I started really looking into it and then moved on to other things. There's quite a lot of interesting stuff one can do with tick data and discrete steps. My emphasis has always been on creating a decent distribution from the data, not on so much on the stats that lead to a trading rule... that said, the bandit problems are neat in that they might get one to start thinking in terms of multiple turn based probabilities, which, actually will lead to the discovery of edges IMO. The 3 days down idea is something along those lines. I'd be interested in anyone posting some decent test results - the 50% result from SP daily closing prices was actually really interesting... maybe in a while I'll post a distribution that may show some better results for that particular time series.
We can't use this to determine if the next closing price will be higher or lower than the current closing price. We can just compare price changes. If the next change will be greater than let's say -30, it could still be -15 and price drops again. Maybe I'm missing something but I can't see how to create a trading rule out of this.