The Annals of Applied Probability

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

Alessandro Arlotto and J. Michael Steele

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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

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

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

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Zentralblatt MATH identifier

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

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


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.

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