Complete Convergence of Short Paths and Karp's Algorithm for the TSP

Let Xi, 1 ≤ i 0 $$\sum^{\infty}_{n=1}p\left|T_{n}/\sqrt{n-\beta}| \gt \epsilon\right \lt \infty.$$ This result is essential in justifying Karp's algorithm for the traveling salesman problem under the independent model, and it settles a question posed by B. W. Weide.