We observe n possibly dependent random variables, the distribution of which is presumed to be stationary even though this might not be true, and we aim at estimating the stationary distribution. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and our estimator. If the dependence within the observations is small, the estimator performs as good as if the data were independent and identically distributed. In addition our estimator is robust to misspecification and contamination. If the dependence is too high but the observed process is mixing, we can select a subset of observations that is almost independent and retrieve results similar to what we have in the i.i.d. case. We apply our procedure to the estimation of the invariant distribution of a diffusion process and to finite state space hidden Markov models.

翻译：我们观测到n个可能相依的随机变量，其分布被假定为平稳的（尽管这一假定可能不成立），并旨在估计该平稳分布。我们建立了目标分布与估计量之间Hellinger距离的非渐近偏差界。当观测内部相关性较弱时，该估计量的表现与数据独立同分布情形下相当。此外，我们的估计量对模型设定错误和污染具有稳健性。若依赖关系过强但观测过程具有混合性，我们可选取一组近似独立的观测子集，并恢复出与独立同分布情形类似的结果。我们将该方法应用于扩散过程不变分布估计及有限状态空间隐马尔可夫模型。