Stability of perpetuities

For a series of randomly discounted terms we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to oo, these being the only possible forms of asymptotic behavior. This solves the existence problem for a one-dimensional perpetuity that remains from a 1979 study by Vervaat, and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one.

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