# On the Koml\'os, Major and Tusn\'ady strong approximation for some classes of random iterates

## TL;DR

This paper extends the Komlós, Major and Tusnádý strong approximation results to functions of random iterates within a Markovian framework, providing new dependent conditions for approximation with rate o(n^{1/p}).

## Contribution

It adapts existing methods to Markovian settings, introducing natural coupling conditions that broaden applicability to various stochastic models.

## Key findings

- Established strong approximation with rate o(n^{1/p}) for functions of random iterates.
- Provided new dependent conditions based on L-infinity or L-1 coupling.
- Demonstrated the optimality of the L-1 coupling condition.

## Abstract

The famous results of Koml\'os, Major and Tusn\'ady (see [15] and [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in L p (p > 2) by a Wiener process with an error term of order o(n 1/p). Very recently, Berkes, Liu and Wu [3] extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in L p is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in L $\infty$ or in L 1), under which the strong approximation result holds with rate o(n 1/p). As we shall see our conditions are well adapted to a large variety of models, including left random walks on GL d (R), contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our L 1-coupling condition is in some sense optimal.