The law of the iterated logarithm for character ratios

Recently, Fulman developed some general connections between martingales and character ratios of a random representation of the symmetric group on transitions, and obtained a convergence rate in a central limit theorem. In this work we aim to establish the law of the iterated logarithm for character ratios. The technique is a well-known Skorokhod embedding theorem for martingales and strong approximation argument. Also, bounded martingale difference methods are used to obtain a large deviation for character ratios.