Random continued fractions: a Markov chain approach

By allowing the numbers appearing in a continued fraction to be random, one gets what are called random continued fractions. Under fairly general conditions, including the case when the random variables are i.i.d. non-negative, random continued fractions converge with probability one. A markovian algorithm seems to play a crucial role in studying the distribution of random continued fractions. This Markov Chain on <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$S=(0,\infty )$</EquationSource> </InlineEquation> is generated by iteration of random monotone decreasing maps on S and the connection comes from the fact that the distribution of random continued fraction is obtained as an invariant probability of the Markov Chain. Using the splitting condition, it is shown that the distribution of the Markov Chain converges exponentially fast in the Kolmogorov distance to an unique invariant probability <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\pi$</EquationSource> </InlineEquation>, which is shown to be non-atomic, except in the degenerate case. A sufficient condition is given for the invariant probability <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\pi$</EquationSource> </InlineEquation> to have full support S. In some special cases, the invaraint probability is obtained explicitly and this includes one case when the probability <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\pi$</EquationSource> </InlineEquation> turns out to be a singular non-atomic probability with full support S. Extensions of some results to higher dimensions are also discussed. Copyright Springer-Verlag Berlin/Heidelberg 2003