Recursive estimation of distributional fix-points

In various stochastic models the random equation S (d) double under bar psi o S of implicit renewal theory appears where the real random variable S an d the stochastic process psi with index space and state space R are indepen dent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of psi. Under integrability assumptions almost sun L-1-convergence is proved. The choice of gains in the recursion is discussed. Applications an given to ins urance mathematics (perpetuities) and queueing theory (stationary waiting a nd queueing times).