## The Annals of Applied Probability

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

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

https://projecteuclid.org/euclid.aoap/1472745454

Digital Object Identifier
doi:10.1214/15-AAP1142

Mathematical Reviews number (MathSciNet)
MR3543892

Zentralblatt MATH identifier
1375.60036

#### 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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