  We establish necessary and sufficient conditions for the shock statistics to
approach self-similar form in Burgers turbulence with L\'{e}vy process initial
data. The proof relies upon an elegant closure theorem of Bertoin and Carraro
and Duchon that reduces the study of shock statistics to Smoluchowski's
coagulation equation with additive kernel, and upon our previous
characterization of the domains of attraction of self-similar solutions for
this equation.
