2011 Putnam 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.

令 $G$ 为具有 $n$ 个元素的阿贝尔群，并令

$$\{g_1=e,g_2,\dots,g_k\} \subsetneqq G$$

是 $G$ 的一组(不一定是最小的)不同生成器。一个特别的

die，随机选择具有相等的元素 $g_1,g_2,...,g_k$ 之一

概率，滚动 $m$ 次，所选元素相乘

产生一个元素$g \in G$。证明存在实数

$b \in (0,1)$ 使得

$$ \lim_{m\to\infty} \frac{1}{b^{2m}} \sum_{x\in G} \left(\mathrm{Prob}(g=x)

- \frac{1}{n}\right)^2 $$

是正的且有限的。