Statistical properties of the attendance time series in the minority game

We study the statistical properties of the attendance time series corresponding to the number of agents making a particular decision in the minority game (MG). We focus on the analysis of the probability distribution and the autocorrelation function of the attendance over a time interval in the efficient phase of the game. In this regime both the probability distribution and the autocorrelation function are shown to have similar behaviour for time differences corresponding to multiples of $2\cdot 2^{m}$, which is twice the number of possible history bit strings in a MG with agents making decisions based on the most recent $m$ outcomes of the game.