TY - JOUR

T1 - Poissonization Principle for a Class of Additive Statistics

AU - Borisov, Igor

AU - Jetpisbaev, Maman

N1 - Funding Information: The study of I. Borisov was supported by the Russian Science Foundation, project no. 22-21-00414. Publisher Copyright: © 2022 by the authors.

PY - 2022/11

Y1 - 2022/11

N2 - In this paper, we consider a class of additive functionals of a finite or countable collection of the group frequencies of an empirical point process that corresponds to, at most, a countable partition of the sample space. Under broad conditions, it is shown that the asymptotic behavior of the distributions of such functionals is similar to the behavior of the distributions of the same functionals of the accompanying Poisson point process. However, the Poisson versions of the additive functionals under consideration, unlike the original ones, have the structure of sums (finite or infinite) of independent random variables that allows us to reduce the asymptotic analysis of the distributions of additive functionals of an empirical point process to classical problems of the theory of summation of independent random variables.

AB - In this paper, we consider a class of additive functionals of a finite or countable collection of the group frequencies of an empirical point process that corresponds to, at most, a countable partition of the sample space. Under broad conditions, it is shown that the asymptotic behavior of the distributions of such functionals is similar to the behavior of the distributions of the same functionals of the accompanying Poisson point process. However, the Poisson versions of the additive functionals under consideration, unlike the original ones, have the structure of sums (finite or infinite) of independent random variables that allows us to reduce the asymptotic analysis of the distributions of additive functionals of an empirical point process to classical problems of the theory of summation of independent random variables.

KW - additive functional

KW - empirical point process

KW - group frequency

KW - Poisson point process

KW - Poissonization

UR - http://www.scopus.com/inward/record.url?scp=85141876805&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/eba1343a-d9f0-3312-8568-66adb2c48b73/

U2 - 10.3390/math10214084

DO - 10.3390/math10214084

M3 - Article

AN - SCOPUS:85141876805

VL - 10

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 21

M1 - 4084

ER -