A simple example: each round, you bet a fraction \(\alpha\) of current wealth on a 50/50 coin flip.
Heads: your bet pays off 300% (you get back 3x what you bet). Tails: you lose your bet.
Five investors face the same coin flips — only their bet fractions differ.
The log-utility investor (\(\alpha = \tfrac{1}{4}\)) almost surely outperforms all others as the number of rounds grows.