A stochastic fixed point equation for weighted minima and maxima

A stochastic fixed point equation for weighted minima and maxima

Given any finite or countable collection of real numbers Tj , j ∈ J , we find all solutions F to the stochastic fixed point equation W d = inf j∈J TjWj , where W and the Wj , j ∈ J , are independent real-valued random variables with distribution F and d = means equality in distribution. The bulk of the necessary analysis is spent on the case when |J | ≥ 2 and all Tj are (strictly) positive. Non...

The Weighted Branching Process and Its Associated Stochastic Fixed Point Equation

Of Minima and Maxima

Let h1, · · · , hn be positive integers. We study new sums m(h1, · · · , hn) = h1−1 ∑ r1=0 · · · hn−1 ∑ rn=0 min { r1 h1 , · · · , rn hn } and M(h1, · · · , hn) = h1−1 ∑ r1=0 · · · hn−1 ∑ rn=0 max { r1 h1 , · · · , rn hn } , the first of which times h1 · · ·hn is the number of lattice points in a pyramid of dimension n + 1. We show that m(h1, · · · , hn) (h1 − 1) · · · (hn − 1) = 1 + ∑ ∅6 =I⊆{1...

Weighted Means in Stochastic Approximation of Minima∗

Weighted averages of Kiefer–Wolfowitz-type procedures, which are driven by larger step lengths than usual, can achieve the optimal rate of convergence. A priori knowledge of a lower bound on the smallest eigenvalue of the Hessian matrix is avoided. The asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a Newton-type adaptive algorithm. Several di...

Vector ultrametric spaces and a fixed point theorem for correspondences

In this paper, vector ultrametric spaces are introduced and a fixed point theorem is given for correspondences. Our main result generalizes a known theorem in ordinary ultrametric spaces.