be the vector of returns from a stock
market. Suppose an ``expert'' suggests a portfolio 
, where does not depend
on 
. This would result in a wealth factor 
. We add this to the stock alternatives, 
.
Show that the new growth rate 
is equal to the old growth rate
It should be noted that true experts use portfolios that use side information and
have some dependence on , and therefore contribute positively to the
growth rate.
be the set of all probability densities on 
for
which is optimal. Show that is a convex set. It may be helpful to use
Theorem 
.