Beardwood-halton-hammersley Theorem for Stationary Ergodic Sequences: Construction of a Counterexample

We construct a stationary ergodic process {X1, X2, X3 . . .} such that each Xt, 1 ≤ t <∞, has the uniform distribution on the unit square and the length Ln of the shortest path through {X1, X2, . . . , Xn} 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 the uniform stationary distribution. Mathematics Subject Classification (2000): Primary 60D05, 90B15; Secondary 60F15, 60G10, 60G55, 90C27.