  We find a nonlinear dependence between an indicator of the degree of
multiscaling of log-price time series of a stock and the average correlation of
the stock with respect to the other stocks traded in the same market. This
result is a robust stylized fact holding for different financial markets. We
investigate this result conditional on the stocks' capitalization and on the
kurtosis of stocks' log-returns in order to search for possible confounding
effects. We show that a linear dependence with the logarithm of the
capitalization and the logarithm of kurtosis does not explain the observed
stylized fact, which we interpret as being originated from a deeper
relationship.
