Limiting behaviour of constrained sums of two variables and the principle of a single big jump

This note studies the asymptotic properties of the variable Zd:=X1d|{X1+X2=d}, as d→∞d→∞. Here X1X1 and X2X2 are non-negative i.i.d. variables with a common twice differentiable density function ff. General results concerning the distributional limits of ZdZd are discussed with various examples.