Statistical Aspects of Perpetuities

For a distribution [mu] on the unit interval we define the associated perpetuity [Psi]([mu]) as the distribution of 1+X1+X1X2+X1X2X3+..., where (Xn)n[set membership, variant] is a sequence of independent random variables with distribution [mu]. Such quantities arise in insurance mathematics and in many other areas. We prove the differentiability of the perpetuity functional[psi] 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.