  In this paper we perform a statistical analysis over the returns and relative
prices of the CAC $40$ and the S\&P $500$ with the purpose of analyzing the
intra-day seasonalities of single and cross-sectional stock dynamics. In order
to do that, we characterized the dynamics of a stock (or a set of stocks) by
the evolution of the moments of its returns (and relative prices) during a
typical day. We show that these intra-day seasonalities are independent of the
size of the bin, and the index we consider, (but characteristic for each index)
for the case of the relative prices but not for the case of the returns.
Finally, we suggest how this bin size independence could be used to
characterize "atypical days" for indexes and "anomalous behaviours" in stocks.
