Section: New Results

Deviation inequalities for martingales with applications

Participant : Xiequan Fan.

In the papers [36] , [37] we study some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett (1962), Freedman (1975), van de Geer (1995), de la Peña (1999) and Pinelis (2006). Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of de la Peña's inequality to self-normalized deviations is also provided.

We also considered an $𝒳$-valued Markov chain $X_1,X_2,...,X_n$ belonging to a class of iterated random functions, which is “one-step contracting" with respect to some distance $d$ on $𝒳$. If $f$ is any separately Lipschitz function with respect to $d$, we use a well known decomposition of $S_n=f(X_1,...,X_n)-𝔼\left[f(X_1,...,X_n)\right]$ into a sum of martingale differences $d_k$ with respect to the natural filtration ${ℱ}_k$. We show that each difference $d_k$ is bounded by a random variable ${\eta }_k$ independent of ${ℱ}_{k-1}$. Using this very strong property, we obtain a large variety of deviation inequalities for $S_n$, which are governed by the distribution of the ${\eta }_k$'s. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain.