 Solution

By Gibbs inequality,with equality exactly when $\boxed{a_j=p_j}$. Put $q_j=u_j/U$, a probability distribution. The optimal log-optimal multiplicative betting rate isFor $U<1$ this is strictly positive by Gibbs inequality; taking $0<\epsilon <g_{\ast }$ in the preceding strong law of large numbers bound proves growth with probability tending to one.

For $U>1$ the casino can set $u_j=U{p}_j$. Every optimal return multiplier is then exactly $1/U$, so the fortune decreases deterministically as $f{U}^{-m}$. More generally $g_{\ast }=-\log U<0$ for that choice. An arbitrary choice with $U>1$ need not cause decline. For example, take $p_1=p_2=1/2$, $u_1=0.01$, $u_2=1.99$. Then $U=2$ but $g_{\ast }=(1/2)\log [0.25/(0.01\cdot 1.99)]>0$. For any fixed positive $p$, making one $u_j$ sufficiently small while preserving $U$ also makes the optimal rate positive.