In this thesis we have analyzed the Auroral Electrojet (AE) Index over the years 2000 to 2005, a time series consisting of over 3 000 000 data points. The aim is to describe this data as a multi-fractal stochastic process. We first introduce a class of random multiplicative measures, which provide the multi-fractality in the stochastic processes that will be defined later. We also review the ...