The expectation operator doesn't care about fatness of tails (well, it kinda doesn't, but note that e.g. the expectation of a random variable with Cauchy distribution is undefined, precisely because of those very fat tails), but the theorem that says that in the long run your wealth is almost always close to its expectation may fail for fat-tail-related reasons.
does maximizing for E(X) lead you to different optimal thetas than maximizing for E(log(X))?
In the present case where we're looking at long-run results only, the answer might be "no" (but -- see above -- I'm not sure it actually is). But in general, if you allow X to be any random variable rather than some kind of long-run average of well-behaved things, it is absolutely not in any way true that maximizing E(X) leads to the same parameter choices as maximizing E(log(X)).
Well, you have to be careful that Kelly Rule assumptions hold.
If you want your choice to be optimal, sure. But all I'm saying is that using "the Kelly rule" to mean "making the choice that maximizes expected log bankroll" seems like a reasonable bit of terminology. Whether using the Kelly rule, in this sense, is a good idea in any given case will of course depend on all sorts of details.
Good point about Cauchy. If even the mean is undefined, all bets are off :-)
it is absolutely not in any way true that maximizing E(X) leads to the same parameter choices as maximizing E(log(X))
Can I get an example? Say, X is a random positive real number. For which distribution which parameters that maximize E(X) will not maximize E(log(X))?
using "the Kelly rule" to mean "making the choice that maximizes expected log bankroll" seems like a reasonable bit of terminology.
I don't know about that. The Kelly Rule means a specific strategy in a specific setting and diluting and fuzzifying that specificity doesn't seem useful.
A lottery ticket sometimes has positive expected value, (a $1 ticket might be expected to pay out $1.30). How many tickets should you buy?
Probably none. Informally, all but the richest players can expect to go broke before they win, despite the positive expected value of a ticket.
In more precise terms: In order to maximize the long-term growth rate of your money (or log money), you'll want to put a very small fraction of your bankroll into lotteries tickets, which will imply an "amount to invest" that is less than the cost of a single ticket, (excluding billionaires). If you put too great a proportion of your resources into a risky but positive expected value asset, the long-term growth rate of your resources can become negative. For an intuitive example, imagine Bill Gates dumping 99% percent of his wealth into a series of positive expected-value bets with single-lottery-ticket-like odds.
This article has some graphs and details on the lottery. This pdf on the Kelly criterion has some examples and general dicussion of this type of problem.
Can we think about Pascal mugging the same way?
The applicability might depend on whether we're trading resource-generating-resources for non-resource-generating assets. So if we're offered something like cash, the lottery ticket model (with payout inversely varying with estimated odds) is a decent fit. But what if we're offered utility in some direct and non-interest-bearing form?
Another limit: For a sufficiency unlikely but positive-expected-value gamble, you can expect the heat death of the universe before actually realizing any of the expected value.