Putnam 2011 A6

Let $G$ be an abelian group with $n$ elements, and let $$\{g_1=e,g_2,\dots,g_k\} \subsetneqq G$$ be a (not necessarily minimal) set of distinct generators of $G$. A special die, which randomly selects one of the elements $g_1,g_2,...,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g \in G$. Prove that there exists a real number $b \in (0,1)$ such that

$$\lim_{m\to\infty} \frac{1}{b^{2m}} \sum_{x\in G} \left(\mathrm{Prob}(g=x) - \frac{1}{n}\right)^2$$ is positive and finite.