  The statistical properties of the increments x(t+T) - x(t) of a financial
time series depend on the time resolution T on which the increments are
considered. A non-parametric approach is used to study the scale dependence of
the empirical distribution of the price increments x(t+T) - x(t) of S&P Index
futures, for time scales T, ranging from a few minutes to a few days using
high-frequency price data. We show that while the variance increases linearly
with the timescale, the kurtosis exhibits anomalous scaling properties,
indicating a departure from the iid hypothesis. Study of the dependence
structure of the increments shows that although the autocorrelation function
decays rapidly to zero in a few minutes, the correlation of their squares
exhibits a slow power law decay with exponent 0.37, indicating persistence in
the scale of fluctuations. We establish a link between the scaling behavior and
the dependence structure of the increments : in particular, the anomalous
scaling of kurtosis may be explained by "long memory" properties of the square
of the increments.
