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Functional Limit Theorem for the Sums of PSI-Processes with Random Intensitiesстатья
Статья опубликована в высокорейтинговом журнале
Статья опубликована в журнале из списка Web of Science и/или Scopus- Авторы: Yakubovich Yuri, Rusakov Oleg, Gushchin Alexander
- Журнал: Mathematics
- Том: 10
- Номер: 21
- Год издания: 2022
- Издательство: MDPI
- Местоположение издательства: Basel, Switzerland
- Первая страница: 1
- Последняя страница: 17
- Номер статьи: 3955
- DOI: 10.3390/math10213955
- Аннотация: We consider a sequence of i.i.d. random variables, (ξ)=(ξ_i)i=0,1,2,⋯, Eξ_0=0, Eξ^2_0=1, and subordinate it by a doubly stochastic Poisson process Π(λt), where λ≥0 is a random variable and Π is a standard Poisson process. The subordinated continuous time process ψ(t)=ξ_{Π(λt)} is known as the PSI-process. Elements of the triplet (Π,λ,(ξ)) are supposed to be independent. For sums of n independent copies of such processes, normalized by \sqrt{n}, we establish a functional limit theorem in the Skorokhod space D[0,T], for any T>0, under the assumption E|ξ_0|^{2h}<∞ for some h>1/γ^2. Here, γ∈(0,1] reflects the tail behavior of the distribution of λ, in particular, γ≡1 when Eλ<∞. The limit process is a stationary Gaussian process with the covariance function Ee^{−λu}, u≥0. As a sample application, we construct a martingale from the PSI-process and establish a convergence of normalized cumulative sums of such i.i.d. martingales.
- Добавил в систему: Гущин Александр Александрович