Convergence of a branching type recursion with non-stationary immigration |
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Authors: | Michael Cramer |
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Institution: | (1) University of Freiburg, Institut für Mathematische Stochastik, Eckerstr. 1, 79104 Freiburg, Germany |
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Abstract: | The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is
given by
, where (X
l
) is a random sequence, (L
n
−1(1)
) are iid copies ofL
n−1,K is a random number andK, (L
n
−1(1)
), {(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
. |
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Keywords: | |
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