Some results concerning the rates of convergence of random walks on finite group (Q1382189)

The following assertion is proved: Let \(P,Q\) be probability measures on a finite group \(G\) of order \(n\) and let \(\| P-U\| \leq 1/m , \| Q - U \| \leq 1/m , \) where \(\|\cdot\| \) denotes the variation, \(m > 2n/(n-1)\), and \(U\) is the uniform distribution on \(G\) . Then for any \( s\in G \), \((P\ast Q)(s) \geq {{m-2}\over m} U(s) .\) The result improves a result by \textit{P. Diaconis} [``Group representations in probability and statistics'' (1988; Zbl 0695.60012)].