Uniform iterated logarithm laws for martingales and their application to functional estimation in controlled Markov chains

In the first part, we establish an upper bound of an iterated logarithm law for a sequence of processes Mn(.)∈C(Rd,Rp) endowed with the uniform convergence on compacts, where Mn(x) is a square integrable martingale for each x in Rd. In the second part we present an iterative kernel estimator of the driving function f of the regression model:Xn+1=f(Xn)+εn+1.Strong convergences and CLT results are proved for this estimator and then extended to controlled Markov models.