Beardwood-Halton-Hammersley Theorem for Stationary Ergodic Sequences: a Counterexample

TL;DR

This paper constructs a stationary ergodic process with uniform marginals where the shortest path length does not follow the classical asymptotic behavior, challenging the extension of the Beardwood-Halton-Hammersley theorem.

Contribution

It provides a counterexample demonstrating that the BHH theorem does not hold for stationary ergodic sequences with uniform marginals.

Findings

Shortest path length $L_n$ is not asymptotic to a constant times $ oot n$.

The classical BHH theorem does not extend to stationary ergodic sequences.

Counterexample shows limitations of existing asymptotic results.

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 theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions.