I will present an easy way to weaken the classical independence condition. Then weakening this condition having in view modeling of time series leaded Murray Rosenblatt and other main actors of probability and statistics to use strong mixing type conditions.
I was working a lot with such conditions until the time I realized it was perhaps not the only way for this. Thus in 1999 I introduced weak dependence conditions with Sana Louhichi. Definitely alternative exist, both for general conditions as in Wei Biao Wu or from the "French" team with Rio, Merlevède and others.
My main purpose is modeling so that I will better turn to provide a wide range of time series.
For those weak dependence conditions a wide range of tools was built with a small french group. Anyway keeping in mind modelling we provide with Xiequan Fan and Dedecker a number of powerful inequalities.
I will quickly describe some of them to switch to some few applications in order to demonstrate that such simple conditions may be somehow productive, for example in the application field of ecology with my project: doukhan.perso.cyu.fr/ecodep.html
Links to underlying papers:
www.sciencedirect.com/science/article/pii/S0304414999000551