Abstract
Exponential tail bounds are derived for solutions of max-recursive equations and for max-recursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise in the worst case analysis of divide and conquer algorithms, in parallel search algorithms or in the height of random tree models. For the proof we determine asymptotic bounds for the moments or for the Laplace transforms and apply a characterization of exponential tail bounds due to Kasahara (1978).
Highlights
Stochastic recursive equations of max-type arise in a great variety of problems with a recursive stochastic component as in the probabilistic analysis of algorithms or in combinatorial optimization problems
Exponential tail bounds are derived for solutions of max-recursive equations and for maxrecursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms
In particular they arise in the worst case analysis of divide and conquer algorithms, in parallel search algorithms or in the height of random tree models
Summary
Stochastic recursive equations of max-type arise in a great variety of problems with a recursive stochastic component as in the probabilistic analysis of algorithms or in combinatorial optimization problems. A general distributional limit theorem for this type of max-recurrences was given in Neininger and Ruschendorf (2005) , see [9], [10] by means of the contraction method. R=1 where (Ar, b) are limits in L2 of the coefficients (Ar(n), br(n)) and Xr are independent copies of X. for some scaling sequences sn have exponential tails. For this case some existence and uniqueness results have been obtained in [8], [10]. We establish various conditions which imply bounds for the asymptotics of moments and Laplace transforms which again lead by Kasahara’s theorem to exponential tail bounds for max-recurrences (Xn). As example we discuss the worst case of FIND sequence
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