We propose a novel approach to concentration for non-independent random variables. The main idea is to ``pretend'' that the random variables are independent and pay a multiplicative price measuring how far they are from actually being independent. This price is encapsulated in the Hellinger integral between the joint and the product of the marginals, which is then upper bounded leveraging tensorisation properties. Our bounds represent a natural generalisation of concentration inequalities in the presence of dependence: we recover exactly the classical bounds (McDiarmid's inequality) when the random variables are independent. Furthermore, in a ``large deviations'' regime, we obtain the same decay in the probability as for the independent case, even when the random variables display non-trivial dependencies. To show this, we consider a number of applications of interest. First, we provide a bound for Markov chains with finite state space. Then, we consider the Simple Symmetric Random Walk, which is a non-contracting Markov chain, and a non-Markovian setting in which the stochastic process depends on its entire past. To conclude, we propose an application to Markov Chain Monte Carlo methods, where our approach leads to an improved lower bound on the minimum burn-in period required to reach a certain accuracy. In all of these settings, we provide a regime of parameters in which our bound fares better than what the state of the art can provide.

翻译：我们针对非独立随机变量提出了一种新颖的集中性方法。主要思想是“假设”随机变量相互独立，并支付一个衡量它们与实际独立程度差距的乘法代价。这一代价通过联合分布与边际分布乘积之间的Hellinger积分来表征，随后利用张量积性质进行上界估计。我们的界限是存在依赖性时集中不等式的自然推广：当随机变量相互独立时，可精确恢复经典结论（如McDiarmid不等式）。此外，在“大偏差”情形下，即使随机变量呈现非平凡依赖关系，我们仍能获得与独立情形相同的概率衰减速度。为验证这一点，我们考虑了若干实际应用：首先，为有限状态空间马尔可夫链提供界限；其次，针对非收缩马尔可夫链——简单对称随机游走，以及依赖完整历史信息的非马尔可夫过程进行分析；最后，我们将该方法应用于马尔可夫链蒙特卡洛方法，得到了达到特定精度所需最小燃烧期的改进下界。在所有场景中，我们均给出了参数区间，证明所提界限优于现有技术水平。