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SOME PROPERTIES OF A RANDOM LINEAR DIFFERENCE EQUATION 1
Author(s) -
Pakes Anthony G.
Publication year - 1983
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1983.tb00388.x
Subject(s) - mathematics , limiting , branching process , limit (mathematics) , asymptotic distribution , distribution (mathematics) , simple (philosophy) , power series , branching (polymer chemistry) , mathematical analysis , statistical physics , combinatorics , statistics , physics , mechanical engineering , philosophy , epistemology , estimator , engineering , materials science , composite material
Summary Some simple conditions are given for the absolute continuity of the limiting distribution of a random linear difference equation. These results are applied to the super‐critical Bellman‐Harris branching process with immigration. When the coefficients of the difference equation are non‐negative and there is no limiting distribution, it is shown that the asymptotic behaviour of the solutions is the same as that of the partial sums of a divergent random power series. A number of limit theorems are given for the latter situation.