Uwe Rösler

The contraction method for recursive algorithms

In this paper we give an introduction to the analysis of algorithms by the contraction method. By means of this method several interesting classes of recursions can be analyzed as particular cases of our general framework. We introduce the main steps of this technique which is based on contraction properties of the algorithm with respect to suitable probability metrics. Typically the limiting distribution is characterized as a fixed point of a limiting operator on the class of probability distributions. We explain this method in the context of several “divide and conquer” algorithms. In the second part of the paper we introduce a new quite general model for branching dynamical systems and explain that the contraction method can be applied in this model. This model includes many classical examples of random trees and gives a general frame for further applications.

A fixed point theorem for distributions

We study in a systematic form the contractive behavior of the map S of distributions to distributions , Xi are independent r.v., L(Xi) = F. Further we show higher and exponential moments of the fixed point. Applications of this structure are given for (a) weighted branching processes, (b) the Hausdorff dimension of random Cantor sets and (c) the sorting algorithm Quicksort.

On the analysis of stochastic divide and conquer algorithms

This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the running time of the algorithm. As a special example we analyse the average performance and the running time distribution of the (2k + 1)-median version of Quicksort.

Martingales with given maxima and terminal distributions

Let μ be any probability measure onR with λ |x|dμ(x)<∞, and let μ* denote its associated Hardy and Littlewood maximal p.m. It is shown that for any p.m.v for which μ<ν<μ* in the usual stochastic order, there is a martingale (Xt)0≦t≦1 for which sup0≦t≦1Xt andX1 have respective p.m. sv and μ. The proof uses induction and weak convergence arguments; in special cases, explicit martingale constructions are given. These results provide a converse to results of Dubins and Gilat [6]; applications are made to give sharp martingale and ‘prophet’ inequalities.

Fixed points with finite variance of a smoothing transformation

Let T=(T1,T2,T3,...) be a sequence of real random variables. We investigate the following fixed point equation for distributions [mu]: W[congruent with][summation operator]j=1[infinity] TjWj, where W,W1,W2,... have distribution [mu] and T,W1,W2,... are independent. The corresponding functional equation is [phi](t)=E [product operator]j=1[infinity] [phi](tTj), where [phi] is a characteristic function. We consider solutions of the fixed point equation with finite variance. Results about existence and uniqueness are derived. In the situation of solutions with zero expectation we give a representation of the characteristic functions of solutions and treat the question of moments and -Lebesgue densities. The article extends results on the case of non-negative T and non-negative solutions.

Stochastic and convex orders and lattices of probability measures, with a martingale interpretation

Letμ be any probability measure on ℝ with ∫|x|dμ(x)<∞ and letμ* denote the associated Hardy and Littlewood maximal p.m., the p.m. of the Hardy and Littlewood maximal function obtained fromμ. Dubins and Gilat [6] showed thatμ* is the least upper bound, in the usual stochastic order, of the collection of p.m.’sν on ℝ for which there is a martingale (Xt)0≤t≤1 having distributions ofX1 and sup0≤t≤1Xt given byμ andν respectively. In this paper, a type of ‘dual representation’ is given. Specifically, letν be any p.m. on ℝ with lim supx→∞xν[x,∞)=0xν[x, ∞)=0 and finitex0=inf{z :ν(−∞,z]0}. Then there is a ‘minimal p.m.’νΔ which is the greatest lower bound, in the usual convex order, of the collection of p.m.’sμ on ℝ for which there is a martingale (Xt)0≤t≤1 having distributions ofX1 and sup0≤t≤1Xt given byμ andν respectively. To demonstrate existence and to obtain identification of these minimal p.m.’s, we use, in particular, a lattice structure on the set of p.m.’s with the convex order, and an equivalence between a convex order of p.m.’s and the stochastic order of their maximal p.m.’s. Consequences of these order results include sharp expectation-based inequalities for martingales. These martingale inequalities form a new class of ‘prophet inequalities’ in the context of optimal stopping theory.

A duality relation for entrance and exit laws for Markov processes

Markov processes Xt on (X, FX) and Yt on (Y, FY) are said to be dual with respect to the function f(x, y) if Exf(Xt, y) = Eyf(x, Yt for all x [epsilon] X, y [epsilon] Y, t [greater-or-equal, slanted] 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1{x

Convergence of the maximum a posteriori path estimator in hidden Markov models

In a hidden Markov model (HMM) the underlying finite-state Markov chain cannot be observed directly but only by an additional process. We are interested in estimating the unknown path of the Markov chain. The most widely used estimator is the maximum a posteriori path estimator (MAP path estimator). It can be calculated effectively by the Viterbi (1967) algorithm as is, e.g., frequently done in the field of coding theory, correction of intersymbol interference, and speech recognition. We investigate (component-wise) convergence of the MAP path estimator. Convergence is shown under the condition of unbounded likelihood ratios. This condition is satisfied in the important case of HMMs with additive white Gaussian noise. We also prove convergence, if the Markov chain has two states. The so-called Viterbi paths are an important tool for obtaining these results.

SOME CONTRIBUTIONS TO CHERNOFF-SAVAGE THEOREMS

An elementary taew proof for the asymptotic normality of simple linear rank statistics is presented which is based on partial integration. Extensions into various directions are possible. Here we consider, as a generalization, weakly dependent processes and improve previous results.

A stochastic fixed point equation for weighted minima and maxima

Given any finite or countable collection of real numbers Tj, j 2 J, we find all solutions F to the stochastic fixed point equation W d = inf j2J T jWj,