On a class of characterization problems for random convex combinations
The authors consider equations of the form \[ X{\overset {d} =}W_1 \cdot X+W_2\cdot X',\tag{1} \] where \(W=(W_1,W_2)\) is a random vector with \(E(W_1+W_2)=1\) and \(X{\overset {d} =}X'\), with \(W,X\) and \(X'\) independent in the right-hand side. Special cases of equations of this type have been considered by several authors. The authors give an existence and uniqueness theorem for solutions of (1) under a moment condition, based on a contraction property and using a special metric. As a very special case they mention a characterization of the exponential distribution related to the total staying time in an M/M/1 queue. There is a result on tail behaviour of the solutions of (1) satisfying a moment condition. Some extensions are mentioned.
scientific article; zbMATH DE number 1310122