Break-down of scaling and convergence to Gaussian distribution in stock market data

We analyze the Standard and Poor's 500 index data of the New York Stock Exchange for more than 32 years. It was suggested earlier that the high frequency data are well described by a truncated Lévy distribution and scaling with respect to the sampling time differences was found. The truncated character of the distribution implies that scaling must break down and that the distribution ultimately converges to a Gaussian. We show by comparing Lévy and Gaussian fits that the characteristic time of the break-down of scaling is of the order of few days. The analysis of the dependence of the kurtosis on the time differences shows that this is much shorter than the time needed for the convergence to the Gaussian being of the order of months.