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August 2016 Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample
Alessandro Arlotto, J. Michael Steele
Ann. Appl. Probab. 26(4): 2141-2168 (August 2016). DOI: 10.1214/15-AAP1142


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.


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Alessandro Arlotto. J. Michael Steele. "Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample." Ann. Appl. Probab. 26 (4) 2141 - 2168, August 2016.


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

zbMATH: 1375.60036
MathSciNet: MR3543892
Digital Object Identifier: 10.1214/15-AAP1142

Primary: 60D05 , 90B15
Secondary: 60F15 , 60G10 , 60G55 , 90C27

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

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.26 • No. 4 • August 2016
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