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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 $L_n$ ofthe shortest path through the points $X_1, X_2, \ldots,X_n$ is not asymptoticto a constant times the square root of $n$. In other words, we show that theBeardwood, Halton and Hammersley theorem does not extend from the case ofindependent uniformly distributed random variables to the case of stationaryergodic sequences with uniform marginal distributions.

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