Exponential Bounds and Tails for Additive Random Recursive Sequences

Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method as well as various characterization results of distributions with exponential tails. The paper generalizes previous results on linear exponential tails and on subgaussian tails to more general classes of additive random recursive sequences. We establish conditions which imply exponential bounds and tail bounds of different strength and order. In particular we give sufficient conditions for tail bounds of the order exp(−atp). The proof of these tail bounds is based on a classical characterization result of Kasahara (1978).