Polynomial tails of additive-type recursions

Eva-Maria Schopp

doi:10.46298/dmtcs.3566

Abstract

Polynomial 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 mainly focuss on polynomial tails that arise due to heavy tail bounds of the toll term and the starting distributions. Besides estimating the tail probability directly we use a modified version of a theorem from regular variation theory. This theorem states that upper bounds on the asymptotic tail probability can be derived from upper bounds of the Laplace―Stieltjes transforms near zero.

Highlights

We are interested in sequences (Xn)n∈N0 satisfying
Xn is the parameter of a problem of size n, which is split into M ≥ 1 subproblems r of random sizes Ir(n) ∈ {0, . . . , n − 1}. (Xn(r)) are distributional copies of (Xn) that correspond to the contribution of subgroup r. bn is a random toll function term and Ar(n) are random factors weighting the subproblems
We prove the assertion for Yk, k ∈ N0, by induction on k

Summary

Fifth Colloquium on Mathematics and Computer Science

Polynomial 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. Besides estimating the tail probability directly we use a modified version of a theorem from regular variation theory. This theorem states that upper bounds on the asymptotic tail probability can be derived from upper bounds of the Laplace–Stieltjes transforms near zero.

Introduction

Xk n

First note that

Integration by