The asymptotic null distribution of the Box-Pierce Q-statistic for random variables with infinite variance an application to German stock returns

The asymptotic null distribution of the Box-Pierce Q-statistic is considered. For random variables with probability distribution in the domain of attraction of a stable law with characteristic exponent α, 1 < α < 2, it is shown that this does not tend to x2 in distribution but, after suitable normalisation, to a rather complicated limit law. The density and selected quantiles of the limit distribution are obtained with numerical methods. In a Monte Carlo experiment the small sample behaviour of the modified test is studied and an empirical application to the 30 most busy German stock returns shows that the significance of observed higher-order correlations is thereby reduced in general.