The average winnings settle near $m<m/2$

Let $X_n$ be the amount that someone wins on a trial $n$, in a game of chance, with $X_i$ independent, each with mean $m<0$ and variance $\sigma^2$. Then, as $n\to\infty$, $P\bigl(\frac{X_1+\dots+X_n}n<\frac m2\bigr)$ will tend to: