maximal inequalities for dependent random variables I want to know literature about maximal inequalities for dependent  random variables i.e. upper bound for $P(\max_{n\ge k\ge 1}\sum_{i=1}^{k}X_i > \delta)$ where $X_i$ are dependent random variables. I am aware of martingale cases, want to know about other scenarios. Thanks. 
 A: If you are interested in non-asymptotic bounds, the following references can be useful (of course, the list is far from being complete).


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*The martingale case is addressed in  Nagaev, S. V. On probability and moment inequalities for supermartingales and martingales. Proceedings of the Eighth Vilnius Conference on Probability Theory and Mathematical Statistics, Part II (2002). The tail function of the maximal of partial sums is expressed in terms of the tail functions of the quadratic variance and the maxima of increments. 

*Probability inequalities for mixing sequences have also been investigated. 
For $\alpha$-mixing sequences, chapter 6 Rio's book Théorie asymptotique des processus aléatoires faiblement dépendants contains a bound of $\mathbb P\left(\max_{n\ge k\ge 1}\sum_{i=1}^{k}X_i > \delta\right)$ using the maxima of quantile functions, the inverse of the sequence of $\alpha$-mixing coefficients and $\sum_{i,j}|\operatorname{Cov}(X_i,X_j)|$. In Shao, Qi Man Maximal inequalities for partial sums of ρ-mixing sequences. Ann. Probab. 23 (1995), no. 2, 948–965, the case of $\rho$-mixing sequences is investigated. 

*A probability inequality under the physical measure of dependence has been found in  Liu, Weidong; Xiao, Han; Wu, Wei Biao Probability and moment inequalities under dependence. Statist. Sinica 23 (2013), no. 3, 1257–1272.
Other dependence structures have also been addressed, like $\tau$ dependence (by Jérôme Dedecker and Clémentine Prieur) or $\beta$-mixing (absolutely regular) sequences (Gabrielle Viennet).
