Statistical Aspects of Perpetuities

For a distribution μ on the unit interval we define the associated perpetuity Ψ(μ) as the distribution of 1+X1+X1X2+X1X2X3+…, where (Xn)n∈N is a sequence of independent random variables with distribution μ. Such quantities arise in insurance mathematics and in many other areas. We prove the differentiability of the perpetuity functionalψ with respect to integral and supremum norms. These results are then used to investigate the statistical properties of empirical perpetuities, including the behaviour of bootstrap confidence regions.