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The theorem says nothing of the sort. By your argument, if I flip a coin 10M times, where heads is 1 and tails is -1, and get a mean of 0, then I will very rarely see tails. You're confusing mean and variance.

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Sorry. I said some pretty silly things.

See the new version of the posts. Hopefully it's a bit more coherent now.

Ok, I think your main idea was that since they've achieved consistently positive returns, then they're likely to keep it up. Even if that were true, the 800lb gorilla question is how they've managed to achieve such consistency in the first place.

They have slightly positive return per trade, and they make millions of trades per day. Their PnL is the sum of their trades -- the expected values grows like O(n), while the volatility grows like O(sqrt(n)), so for n = several million, the probability their PnL will be negative is extraordinarily low.

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