Suppose we have the wealth process of a gambler, . Such a process is common in, e.g., game-theoretic statistics, information theory, and portfolio optimization.
We are usually interested in making the wealth grow over time (eg in game-theoretic hypothesis testing). A natural question is what sort of criteria should we use to measure the growth?
Often one chooses bets in order to maximize the log-wealth, i.e., the expected value . This is also called Kelly betting.
There are several reasons one might choose to do this:
- In 1956, Kelly pointed out that logarithmic returns add, hence the SLLN applies which makes it easier to reason about the behavior of the log-wealth over time.
- In 1961, Breiman showed that in iid settings, maximizing the log-wealth leads to the reaching a desired threshold ( say, for hypothesis testing purposes) as fast as possible, without risking all of your wealth at any time (which can happen if one maximizes say).