In this talk we will consider random walks with bounded range on regular languages over a finite alphabet. The random walks can be described by a finite set of parameters. The main goal of this talk is to show that drift and asymptotic entropy of transient random walks vary real-analytically in terms of probability measures of constant support. For this purpose, I will give a brief introduction to random walks on regular languages and I will give formulas for the drift and entropy, from which one can deduce analyticity. The main idea is to cut the random walk into pieces such that this sequence of pieces form a hidden Markov chain, from which we get the analytic behaviour of the entropy. Finally, I will consider positive recurrent random walks and I will show how to compute the entropy in that case explicitly.