SECOND-ORDER RENEWAL STRUCTURE IN SUBCRITICAL BELLMAN-HARRIS PROCESSES WITH HEAVY-TAILED LIFETIMES
Ключевые слова:
Bellman–Harris process, age-dependent branching, defective renewal equation, regular variation, subexponential density, convolution defect, Karamata Tauberian theorem, second-order asymptotics.Аннотация
For a subcritical Bellman–Harris process with offspring mean ( ) 0,1 m and a lifetime law
whose tail ( ) ( ) G t t L t − = is regularly varying, we obtain second-order asymptotic expansion of the mean
population size ( ) ( ) A t Z t = in the finite-mean regime 1 . The leading term ( ) ( ) 1 G t m− is classical. For
1 the correction is shown to live at the scale of the lifetime density and to equal, ( ) ( )2 2 1 G
m g t m −
where G
is the mean lifetime. The proofs separate a renewal-theoretic mechanism from a distributional
one. On the renewal side we establish an exact two-term transfer law for the defective renewal operator
generated by mG: its finite mass reproduces the forcing term, while its regularly varying tail contributes a
further term at the density scale, weighted by the total mass of the forcing term. On the distributional side the
mean defect is driven by the convolution defect 2 2 , G G − Above the threshold that defect carries total mass
zero, which removes the second term of the transfer law from the expansion of the mean.
