BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

On strong approximations for some classes of random iterates Merlevède, Florence


his talk is devoted to strong approximations in the dependent setting. The famous results of Koml\'os, Major and Tusn\'ady (1975-1976) state that it is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random variables in ${\mathbb L}^p$ ($p >2$) by a Wiener process with an error term of order $o(n^{1/p})$. In the case of functions of random iterates generated by an iid sequence, we we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb L}^\infty$ or in ${\mathbb L}^1$), under which the strong approximation result holds with rate $o(n^{1/p})$. The proof is an adaptation of the recent construction given in Berkes, Liu and Wu (2014). As we shall see our conditions are well adapted to a large variety of models, including left random walks on $GL_d({\mathbb R})$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We shall also provide some examples showing that our ${\mathbb L}^1$-coupling condition is in some sense optimal. This talk is based on a joint work with J. Dedecker and C. Cuny.

Item Media

Item Citations and Data


Attribution-NonCommercial-NoDerivatives 4.0 International