  We give a short proof of a recent result of Claesson, Dukes, Frankl\'in and
Stef\'ansson, that connects tournament score sequences and the
Erd\H{o}s-Ginzburg-Ziv numbers from additive number theory. We show that this
connection is, in fact, an instance of the L\'evy-Khintchine formula from
probability theory, and highlight how such formulas can be useful in
enumerative combinatorics. Our proof combines renewal theory with the
representation of score sequences as lattice paths, due to Erd\H{o}s and Moser
in the 1960s. These probabilistic and geometric points of view lead to a
simpler proof. A key idea in the original proof and ours is to consider cyclic
shifts of score sequences. We observe, however, that this idea is already
present in Kleitman's remarks added to one of Moser's final articles in 1968,
and in subsequent works by Kleitman. In the same article, Moser conjectured
that there are asymptotically $C4^n/n^{5/2}$ many score sequences of length
$n$. Combining the arguments in the current work with those in a recent work by
the third author, we demonstrate the utility of the L\'evy-Khintchine method,
by giving a short proof of Moser's conjecture.
