ISSN 1055-1344, Siberian Advances in Mathematics, 2018, Vol. 28, No. 2, pp. 115–153.
Allerton Press, Inc., 2018.
Original Russian Text
V.A. Topchi˘ı, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 139–192.
On Renewal Matrices Connected with Branching Processes with
Tails of Distributions of Diﬀerent Orders
V. A. To pchi
Sobolev Institute of Mathematics, Omsk Division, Omsk, 644099 Russia.
Received June 25, 2016
Abstract—We study irreducible renewal matrices generated by matrices whose rows are propor-
tional to various distribution functions. Such matrices arise in studies of multi-dimensional critical
Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are
based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching
processes, unsolved problems are known that correspond to the case in which the tails of some of
the above mentioned distribution functions are integrable, while the other distributions lack this
property. We assume that the heaviest tails are regularly varying at the inﬁnity with parameter −β ∈
[−1, 0) and asymptotically proportional, while the other tails are inﬁnitesimal with respect to them.
Under a series of additional conditions, we describe asymptotic properties of the ﬁrst and second
order increments for the renewal matrices.
Keywords: renewal matrix and its increment, asymptotic representations, regularly varying
functions, Bellman–Harris critical processes.
denote the indicator function of a set A, i.e., the function assuming the value 1 on
each element of A and the value 0 on each element of
A.LetI = I
denote the identity
matrix of order n,whereδ
is the Kronecker delta. Put I(t):=1
, the convolution
C(t)=A ∗ B(t)=
is determined by the relations
Notice that this deﬁnition is well deﬁned for functions of bounded variation which will be considered
We denote the convolution of a family of functions W
(t):=I(t).ForamatrixM(t) that vanishes on the negative half-axis, by the renewal matrix
we mean the matrix