On the Stochastic Euclidean Travelling Salesperson Problem for Distributions with Unbounded Support

We study the asymptotic behavior of the shortest tour Tn through n points X1, ', Xn in ℝdd ≥ 2, where Xii≥1 are i.i.d. random variables, whose density fx has an unbounded support. Beardwood et al. [2] conjectured that Tn/n1-1/d converges a.s. if and only if ∫ fx1-1/ddx < ∞ and ∫ fx‖x‖d/d-1dx < ∞. We disprove this conjecture, and we show that the second integrability condition is not strong enough. We give a stronger condition that is optimal but not necessary.