Thin tails of fixed points of the nonhomogeneous smoothing transform

الأصل · EN

For a given random sequence (C,T₁,T₂,) with nonzero C and a.s. finite number of nonzero Tₖ, the nonhomogeneous smoothing transform S maps the law of a real random variable X to the law of ∑ₖ≥ ₁TₖXₖ+C, where X₁,X₂, are independent copies of X and also independent of (C,T₁,T₂,). This law is a fixed point of S if the stochastic fixed-point equation (SFPE) Xd=∑ₖ≥ ₁TₖXₖ+C holds true, where d= denotes equality in law. Under suitable conditions including E C=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T₁,T₂,) such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.