On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

The famous results of Komlós, Major and Tusnády (see Komlós et al., 1976 [15] and Major, 1976 [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in Lp (p>2) by a Wiener process with an error term of order o(n1∕p). Very recently, Berkes et al. (2014) 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 Lp 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∞ or in L1), under which the strong approximation result holds with rate o(n1∕p). As we shall see our conditions are well adapted to a large variety of models, including left random walks on GLd(R), contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our L1-coupling condition is in some sense optimal.