MathDB
Putnam 2011 A6

Source:

December 5, 2011
Putnamabstract algebraprobabilitylimitreal analysislinear algebramatrix

Problem Statement

Let GG be an abelian group with nn elements, and let {g1=e,g2,…,gk}⊊G\{g_1=e,g_2,\dots,g_k\}\subsetneq G be a (not necessarily minimal) set of distinct generators of G.G. A special die, which randomly selects one of the elements g1,g2,…,gkg_1,g_2,\dots,g_k with equal probability, is rolled mm times and the selected elements are multiplied to produce an element g∈G.g\in G.
Prove that there exists a real number b∈(0,1)b\in(0,1) such that lim⁡m→∞1b2m∑x∈G(Prob(g=x)−1n)2\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2 is positive and finite.
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