Convergence of a branching type recursion with non-stationary immigration

The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by Ln =d Σ^Ki=1 XiL⁽ⁱ⁾n–1 + Yn, where (Xi) is a random sequence, (L⁽ⁱ⁾n–1) are independent, identically distributed copies of Ln–1, K is a random number and K, (L⁽ⁱ⁾n–1), {(Xi), Yn} are independent. This recursion has been studied intensively in the literature in the case that Xi ≥ 0, K is nonrandom and Yn = 0 ∀n. Cramer and Rüschendorf treat the above recursion without immigration with starting condition L0 = 1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution of Yn does not depend on the time n). In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved random variables. The treatment of the recursion is based on a contraction method. The conditions of the limit theorem are built upon the knowledge of the first two moments of Ln. In case of stationary immigration a detailed analysis of the first two moments of Ln leads one to consider 15 different cases. These cases are illustrated graphically and provide a straightforward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalized Ln.