Chernoff's theorem in the branching random walk

If F n∗ is the n -fold Stieltjes convolution of the increasing function F , then a version of Chernoff's theorem, on the limiting behaviour of ( F n∗ ( na )) 1/ n , is established for F n∗ . If Z ( n ) ( t ) is the number of the n th-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z ( n ) .