Betting systems and independence

This is a question about a problem (not homework) from Ed Thorp's book, http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/ElementaryProbability.pdf [Broken]. Problem 13 on page 85 outlines a proof that betting systems designed to make an unfavorable game favorable cannot work when there is a maximum bet limit. Problem 14 outlines an even stronger version. Right from the start, I get lost when he talks about the independence of bets:

Elementary Probability said:
5.13 Failure of the classical gambling systems. A bet in a gambling game is a random variable. Most (but not all) of the standard gambling games consist of repeated independent trials, which means that the bets Bi are independent. Further, there is a constant K such that |Bi| <= K for all i.

I know that the outcomes, win or lose, are assumed to be independent for the game in question (betting on red at roulette, for example). But the bet size for a given trial generally depends on the outcomes of the earlier trials: that is the whole idea of a money management system. So if $$\epsilon_i$$, which takes on values of +1 or -1, is the random variable representing the outcome on the ith trial, and $$W_i(\epsilon_1, ...,\epsilon_{i-1})$$ is the amount wagered on that trial, then in Thorp's notation the random variable for the bet is

$$B_i = \epsilon_i W_i$$

So Bi and Bj are not usually independent. In the special case that the probability of success for each trial is 1/2, the covariance of Bi and Bj would be zero, but we are interested only in games where the expected value is negative for each trial.

Did I simply misunderstand Thorp's notation, or have am I making a conceptual error?

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mathman
There looks like there is some confusion between the outcomes of the bets (independent random variables) and the sizes of the bets, which can be determined by the bettor anyway he chooses.

There looks like there is some confusion between the outcomes of the bets (independent random variables) and the sizes of the bets, which can be determined by the bettor anyway he chooses.

There is, but is it on my part or the book's part?

mathman
I believe the book is talking about outcomes, which are independent. The size limit may be due to the fact that the better has only a finite amount of money.

Yes, it is true that the bettor would start out with a finite amount. But I was interpreting K as the the maximum bet allowed, a limit which applies regardless of how much money the bettor has. That way, the law of large numbers result would apply even if the player had enough money to continue playing (and, with probability 1, losing) forever. It looks like the total winnings after n rounds of betting is

$$S_n = \sum_{i=1}^nB_i$$

So the B_i must include the bet sizes as well as the outcome, positive or negative. The law of large numbers usually applies to independent variables, so I think that's why Thorp says the B_i are independent. But I don't see how they could be.

EDIT:

There is a link to the pdf for the book at the top of my first post, in case you need to read the rest of the problem for more context. It's on p. 85.

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mathman