The Annals of Applied Probability

Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample

Alessandro Arlotto and J. Michael Steele

Full-text: Open access

Abstract

We construct a stationary ergodic process $X_{1},X_{2},\ldots$ such that each $X_{t}$ has the uniform distribution on the unit square and the length $L_{n}$ of the shortest path through the points $X_{1},X_{2},\ldots,X_{n}$ is not asymptotic to a constant times the square root of $n$. In other words, we show that the Beardwood, Halton, and Hammersley [Proc. Cambridge Philos. Soc. 55 (1959) 299–327] theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2141-2168.

Dates
Received: January 2014
Revised: December 2014
First available in Project Euclid: 1 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1472745454

Digital Object Identifier
doi:10.1214/15-AAP1142

Mathematical Reviews number (MathSciNet)
MR3543892

Zentralblatt MATH identifier
1375.60036

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 90B15: Network models, stochastic
Secondary: 60F15: Strong theorems 60G10: Stationary processes 60G55: Point processes 90C27: Combinatorial optimization

Keywords
Traveling salesman problem Beardwood–Halton–Hammersley theorem subadditive Euclidean functional stationary ergodic processes equidistribution construction of stationary processes

Citation

Arlotto, Alessandro; Steele, J. Michael. Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample. Ann. Appl. Probab. 26 (2016), no. 4, 2141--2168. doi:10.1214/15-AAP1142. https://projecteuclid.org/euclid.aoap/1472745454


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