60H25 Random operators and equations [See also 47B80]
- A perpetuity is a real valued random variable which is characterised by a distributional fixed-point equation of the form X=AX+b, where (A,b) is a vector of random variables independent of X, whereas dependencies between A and b are allowed. Conditions for existence and uniqueness of solutions of such fixed-point equations are known, as is the tail behaviour for most cases. In this work, we look at the central area and develop an algorithm to approximate the distribution function and possibly density of a large class of such perpetuities. For one specific example from the probabilistic analysis of algorithms, the algorithm is implemented and explicit error bounds for this approximation are given. At last, we look at some examples, where the densities or at least some properties are known to compare the theoretical error bounds to the actual error of the approximation. The algorithm used here is based on a method which was developed for another class of fixed-point equations. While adapting to this case, a considerable improvement was found, which can be translated to the original method.