Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities

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Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities. / Yakubovich, Yuri ; Rusakov, Oleg; Gushchin, Alexander.

In: Mathematics, Vol. 10, No. 21, 3955, 11.2022.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{f678c7d3ac534c1a91d11fa9f270981e,

title = "Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities",

abstract = "We consider a sequence of i.i.d. random variables, (Formula presented.), (Formula presented.), (Formula presented.), and subordinate it by a doubly stochastic Poisson process (Formula presented.), where (Formula presented.) is a random variable and (Formula presented.) is a standard Poisson process. The subordinated continuous time process (Formula presented.) is known as the PSI-process. Elements of the triplet (Formula presented.) are supposed to be independent. For sums of n, independent copies of such processes, normalized by (Formula presented.), we establish a functional limit theorem in the Skorokhod space (Formula presented.), for any (Formula presented.), under the assumption (Formula presented.) for some (Formula presented.). Here, (Formula presented.) reflects the tail behavior of the distribution of (Formula presented.), in particular, (Formula presented.) when (Formula presented.). The limit process is a stationary Gaussian process with the covariance function (Formula presented.), (Formula presented.). 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.",

keywords = "functional limit theorem, Poisson stochastic index process, pseudo-Poisson process, random intensity",

author = "Yuri Yakubovich and Oleg Rusakov and Alexander Gushchin",

note = "Publisher Copyright: {\textcopyright} 2022 by the authors.",

year = "2022",

month = nov,

doi = "10.3390/math10213955",

language = "English",

volume = "10",

journal = "Mathematics",

issn = "2227-7390",

publisher = "MDPI AG",

number = "21",

}

RIS

TY - JOUR

T1 - Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities

AU - Yakubovich, Yuri

AU - Rusakov, Oleg

AU - Gushchin, Alexander

PY - 2022/11

Y1 - 2022/11

N2 - We consider a sequence of i.i.d. random variables, (Formula presented.), (Formula presented.), (Formula presented.), and subordinate it by a doubly stochastic Poisson process (Formula presented.), where (Formula presented.) is a random variable and (Formula presented.) is a standard Poisson process. The subordinated continuous time process (Formula presented.) is known as the PSI-process. Elements of the triplet (Formula presented.) are supposed to be independent. For sums of n, independent copies of such processes, normalized by (Formula presented.), we establish a functional limit theorem in the Skorokhod space (Formula presented.), for any (Formula presented.), under the assumption (Formula presented.) for some (Formula presented.). Here, (Formula presented.) reflects the tail behavior of the distribution of (Formula presented.), in particular, (Formula presented.) when (Formula presented.). The limit process is a stationary Gaussian process with the covariance function (Formula presented.), (Formula presented.). 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.

AB - We consider a sequence of i.i.d. random variables, (Formula presented.), (Formula presented.), (Formula presented.), and subordinate it by a doubly stochastic Poisson process (Formula presented.), where (Formula presented.) is a random variable and (Formula presented.) is a standard Poisson process. The subordinated continuous time process (Formula presented.) is known as the PSI-process. Elements of the triplet (Formula presented.) are supposed to be independent. For sums of n, independent copies of such processes, normalized by (Formula presented.), we establish a functional limit theorem in the Skorokhod space (Formula presented.), for any (Formula presented.), under the assumption (Formula presented.) for some (Formula presented.). Here, (Formula presented.) reflects the tail behavior of the distribution of (Formula presented.), in particular, (Formula presented.) when (Formula presented.). The limit process is a stationary Gaussian process with the covariance function (Formula presented.), (Formula presented.). 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.

KW - functional limit theorem

KW - Poisson stochastic index process

KW - pseudo-Poisson process

KW - random intensity

UR - http://www.scopus.com/inward/record.url?scp=85141882210&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/11701af3-b2fa-3e90-8657-d2d9fb9d340a/

U2 - 10.3390/math10213955

DO - 10.3390/math10213955

M3 - Article

AN - SCOPUS:85141882210

VL - 10

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 21

M1 - 3955

ER -

ID: 100798597