Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree

• Published in: Lithuanian mathematical journal Vol. 62; no. 4; pp. 447 - 466
• Main Authors: Iksanov, Alexander, Marynych, Alexander, Rashytov, Bohdan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2022; Springer Nature B.V
• Subjects: Actuarial Sciences; Mathematics; Mathematics and Statistics; Number Theory; Ordinary Differential Equations; Probability Theory and Stochastic Processes; Random walk; Stochastic processes; general branching process; secondary 60K05; primary 60F05, 60J80; perturbed random walk; weak convergence; stable Lévy process
• ISSN: 0363-1672; 1573-8825
• DOI: 10.1007/s10986-022-09574-9

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk
η
1
,
ξ
1
 + 
η
2
,
ξ
1
 + 
ξ
2
 + 
η
3
, …, where (
ξ
1
, η
1
)
,
(
ξ
2
, η
2
)
,
… are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by
N
j
(
t
) the number of the
j
th generation individuals with birth times ⩽
t
. Assume that
j
=
j
(
t
) → ∞ and
j
(
t
) =
o
(
t
a
) as
t
→ ∞ for some explicitly given
a >
0 (to be specified in the paper). The corresponding
j
th generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (
N
⌊
j
(
t
)
u
⌋
(
t
))
u
 > 0
, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean.