A limit theorem for random coverings of a circle

LetN _{α, m} equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that lim_α→0 P(N_α,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(N_α,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.