In this article a probability density function and dependence degree analysis of financial time series, namely the Dow Jones and NYSE, is presented. The present study, which aims to give theoretical support to some stylized empirical evidence, is performed under the present non-extensive framework for which the probability distributions that optimize its fundamental information measure form,  , are also the (stationary) solutions of a nonlinear Fokker-Plank equation. One determines the rescaled coefficient of the drift force and diffusion coefficient for both market indices and various aggregated times. Using a generalized form of Kullback-Leibler mutual information, I_q, one analyses the non-Gaussianity of returns using the dependence between stock market index values. The same mutual information form is used to determine the degree of dependence between returns. The analysis shows that this dependence can be considered independent from the time distance τ result that is connected with the long-range correlation in volatility.