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We consider solutions to so-called stochastic fixed point equation , where Ψ is a random Lipschitz function and R is a random variable independent of Ψ. Under the assumption that Ψ can be approximated by the function , we show that the tail of R is comparable with the one of A, provided that the distribution of is tail equivalent. In particular, we obtain new results for the random difference equation. [ABSTRACT FROM AUTHOR]

For a given random sequence ( C , T 1 , T 2 , … ) , the smoothing transform S maps the law of a real random variable X to the law of ∑ k ≥ 1 T k X k + C , where X 1 , X 2 , … are independent copies of X and also independent of ( C , T 1 , T 2 , … ) . This law is a fixed point of S if X = d ∑ k ≥ 1 T k X k + 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 because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on ( C , T 1 , T 2 , … ) such that the canonical solution exhibits right and/or left Poissonian 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. [ABSTRACT FROM AUTHOR]