Asymptotics of the Moments of Extreme-Value Related Distribution Functions

Abstract

The asymptotic cost of many algorithms and combinatorial structures is related to the extremevalue Gumbel distribution exp(− exp(−x)). The following list is not exhaustive: Trie, Digital Search Tree, Leader Election, Adaptive Sampling, Counting Algorithms, trees related to the Register Function, Composition of Integers, some structures represented by Markov chains (Column-Convex Polyominoes, Carlitz Compositions), Runs and number of distinct values of some multiplicity in sequences of geometrically distributed random variables. Sometimes we can start from an exact (discrete) probability distribution, sometimes from an asymptotic analysis of the discrete objects (e.g., urn models) before establishing the relationship with the Gumbel distribution function. Also some Markov chains are either exactly and directly given by the structure itself, or as a limiting Markov process.The main motivation of the paper is to compute the asymptotic distribution and the moments of the random variables in question. The moments are usually given by a dominant part and a small fluctuating part. We use Laplace and Mellin transforms and singularity analysis and aim for a unified treatment in all cases. Furthermore, our goal is a purely mechanical computation of dominant and fluctuating components, with the help of a computer algebra system. We provide each time the first three moments, but the treatment is (almost) completely automatic. We need some real analysis for the approximations and apart from that only easy complex analysis; simple poles and a few special functions.

The asymmetric leader election algorithm: Number of survivors near the end of the game

“…The error term is O(n −C ). We proceed as follows (see [13] for detailed proofs): from the asymptotic properties of the urns, we obtain the asymptotic distributions of our random variables of interest. Next we compute the Laplace transform φ(α) of these distributions, from which we can derive the dominant part of probabilities as well as the (tiny) periodic part in the form of a Fourier series.…”

“…This type of convergence is not uncommon in the Analysis of Algorithms. Many examples are given in [13].…”

“…Thus, the present paper represents another application of the powerful technique that was first presented in [13].…”

“…Following again the techniques detailed in [13], this leads to (again, we have the elimination of p χt .) Theorem 3.2 The periodic component of the probability G(n, J(n) − κ, ) that L: the number of survivors at step J(n) − κ, is given by , (starting with n candidates), is asymptotically given by…”

Advancing in the presence of a demon

“…We are able to compute all moments (asymptotically) in an almost automatic fashion. This will be done with the techniques worked out in [7]. Note that the expectated value for the symmetric case p = q = 1 2 was computed using Rice's method in [10].…”