Moment convergence in renewal theory

A. Iksanov, A. Marynych, M. Meiners

Research output: Book/ReportReportAcademic

Abstract

Let ¿1, ¿2, . . . be independent copies of a positive random variable ¿, and let Sk := ¿ 1 + . . . + ¿ k, k ¿ N0. Define N(t) := #{k ¿ N0 : Sk= t}. (N(t))t=0 is a renewal counting process. It is known that if ¿ is in the domain of attraction of a stable law of index a ¿ (1, 2], then N(t), suitably shifted and scaled, converges in distribution as t ¿ 8 to a random variable with a stable law. We show that in this situation, also the first absolute moments converge to the first absolute moment of the limiting random variable. Further, the corresponding result for subordinators is established.
Original languageEnglish
Publishers.n.
Number of pages7
Publication statusPublished - 2012

Publication series

NamearXiv.org
Volume1208.3964 [math.PR]

Fingerprint

Renewal Theory
Stable Laws
Random variable
Moment
Converge
Subordinator
Counting Process
Renewal Process
Limiting

Cite this

Iksanov, A., Marynych, A., & Meiners, M. (2012). Moment convergence in renewal theory. (arXiv.org; Vol. 1208.3964 [math.PR]). s.n.
Iksanov, A. ; Marynych, A. ; Meiners, M. / Moment convergence in renewal theory. s.n., 2012. 7 p. (arXiv.org).
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Iksanov, A, Marynych, A & Meiners, M 2012, Moment convergence in renewal theory. arXiv.org, vol. 1208.3964 [math.PR], s.n.

Moment convergence in renewal theory. / Iksanov, A.; Marynych, A.; Meiners, M.

s.n., 2012. 7 p. (arXiv.org; Vol. 1208.3964 [math.PR]).

Research output: Book/ReportReportAcademic

TY - BOOK

T1 - Moment convergence in renewal theory

AU - Iksanov, A.

AU - Marynych, A.

AU - Meiners, M.

PY - 2012

Y1 - 2012

N2 - Let ¿1, ¿2, . . . be independent copies of a positive random variable ¿, and let Sk := ¿ 1 + . . . + ¿ k, k ¿ N0. Define N(t) := #{k ¿ N0 : Sk= t}. (N(t))t=0 is a renewal counting process. It is known that if ¿ is in the domain of attraction of a stable law of index a ¿ (1, 2], then N(t), suitably shifted and scaled, converges in distribution as t ¿ 8 to a random variable with a stable law. We show that in this situation, also the first absolute moments converge to the first absolute moment of the limiting random variable. Further, the corresponding result for subordinators is established.

AB - Let ¿1, ¿2, . . . be independent copies of a positive random variable ¿, and let Sk := ¿ 1 + . . . + ¿ k, k ¿ N0. Define N(t) := #{k ¿ N0 : Sk= t}. (N(t))t=0 is a renewal counting process. It is known that if ¿ is in the domain of attraction of a stable law of index a ¿ (1, 2], then N(t), suitably shifted and scaled, converges in distribution as t ¿ 8 to a random variable with a stable law. We show that in this situation, also the first absolute moments converge to the first absolute moment of the limiting random variable. Further, the corresponding result for subordinators is established.

M3 - Report

T3 - arXiv.org

BT - Moment convergence in renewal theory

PB - s.n.

ER -

Iksanov A, Marynych A, Meiners M. Moment convergence in renewal theory. s.n., 2012. 7 p. (arXiv.org).