Convergence of a branching type recursion with non-stationary immigration

Abstract

The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by$L_n = d\sum _{i - 1}^K X_1 L_{n - 1}^{(1)} + Y_n ,$, where (X _l ) is a random sequence, (L ⁻¹⁽¹⁾ _n ) are iid copies ofL _n−1,K is a random number andK, (L ⁻¹⁽¹⁾ _n ), {(X _l ),Y _n } are independent.

This recursion has been studied intensively in the literature in the case thatX _l ≥0,K is nonrandom andY _n =0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL ₀=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY _n does not depend on the timen).

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 ofL _n . In case of stationary immigration a detailed analysis of the first two moments ofL _n leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL _n .