Covering the circle with random ARCS

AbstractArcs of lengthsln, 0<ln+1<=ln<1,n=1,2,…, are thrown independently and uniformly on a circumferenceC of unit length. The union of the arcs coversC with probability one if and only if $$\sum\limits_{n = 1}^\infty {n^{ - 2} \exp \left( {l_1 + ... + 1l_n } \right) = \infty } $$ .