[PairTheBoard]

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BillC --
<font color="white"> ,
</font> The expected value of the bankroll is a decreasing function of time. It has a negative derivative, i.e., a negative growth rate.



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FALSE

Let me give a simple example that proves you are wrong here Bill.

Consider a weighted coinflip where you have a 55% chance of winning. Regular 1-1 payoff. Let your initial bankroll be 1 unit and make one coinflip bet per unit of time. Your proportional betting will be 100% of your current bankroll.

What is the expected value of your bankroll at time t=n?

E[Bn] = (.55)^n * 2^n = 1.1^n

The "expected value of your bankroll as a function of time" not only is NOT decreasing, it is an Increasing Function of Time. In fact, the expected value of your bankroll as a function of time Grows Exponentially to Infinity.

This is true even though the "Growth Rate" for your Bankroll is Negative and your Bankroll at time t=n converges in probabilty to zero as n--&gt;infinity.

PairTheBoard