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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.

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  • $\begingroup$ There is the ergodic maximal theorem if $(X_n)$ is stationary and $X_n\in L^1$. This says $\mathbb P(\sup_n \frac 1n\sum_{i=1}^n|X_i|>\delta)<\mathbb E|X_1|/\delta$. $\endgroup$
    – Anthony Quas
    Commented Sep 19, 2015 at 12:50
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    $\begingroup$ "negative association" or "negative dependence", a case where conditioning on $X_i$ being large decreases the chance of all $X_j$ being large. $\endgroup$
    – usul
    Commented Sep 20, 2015 at 0:17

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If you are interested in non-asymptotic bounds, the following references can be useful (of course, the list is far from being complete).

  • 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).

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