Ufa Mathematical Journal
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Ufimsk. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Ufa Mathematical Journal, 2021, Volume 13, Issue 4, Pages 123–130
DOI: https://doi.org/10.13108/2021-13-4-123
(Mi ufa596)
 

This article is cited in 1 scientific paper (total in 1 paper)

Law of large numbers for weakly dependent random variables with values in $D\left[0,1\right]$

O. Sh. Sharipova, A. F. Norjigitovb

a National University of Uzbekistan named after Mirzo Ulugbek, 100174 University street 4, Almazar district, Tashkent, Uzbekistan
b V.I. Romanovskiy Institute of Mathematics, Academy of Sciences of Uzbekistan, 100174 University street 4b, Almazar district, Tashkent, Uzbekistan
References:
Abstract: Limit theorems in Banach spaces are important, in particular, because of applications in functional data analysis. This paper is devoted to the law of large numbers for the random variables with values in the space $D\left[0,1\right]$. This space is not separable if we consider it with supremum norm and it is difficult to prove limit theorems in this space. The law of large numbers is well-studied for the sequences of independent $D\left[0,1\right]$-valued random variables. It is known that in the case of independent and identically distributed random variables with values in $D\left[0,1\right]$ the existence of the first moment of the norm of random functions is a necessary and sufficient condition for the strong law of large numbers. The law of large numbers for the sequences of independent and not necessarily identically distributed random variables with values in $D\left[0,1\right]$ were proved as well. Our main goal is to prove the law of large numbers for the weakly dependent random variables with values in the space $D\left[0,1\right]$. Namely, we consider the sequences of mixing random variables with values in $D\left[0,1\right]$. Mixing conditions for $D\left[0,1\right]$-valued random variables can be introduced in several ways. One can assume that random functions themselves satisfy mixing conditions. We consider a slightly different condition. In fact we assume that the finite dimensional projections of the $D[0,1]$-valued random variables satisfy mixing condition. This is a weaker condition than assuming that random functions themselves satisfy mixing condition. In the paper the law of large numbers for $\rho_{m}$-mixing sequences of $D\left[0,1\right]$-valued random variables are proved.
Keywords: Law of large numbers, mixing sequence, $D\left[0,1\right]$ space.
Funding agency
The research of the first author was supported by the Collaborative Research Grant SFB 823 “Statistical modelling of nonlinear dynamic processes” (Germany).
Received: 27.09.2020
Russian version:
Ufimskii Matematicheskii Zhurnal, 2021, Volume 13, Issue 4, Pages 126–133
Bibliographic databases:
Document Type: Article
UDC: 519.21
MSC: 60F05, 60D05
Language: English
Original paper language: English
Citation: O. Sh. Sharipov, A. F. Norjigitov, “Law of large numbers for weakly dependent random variables with values in $D\left[0,1\right]$”, Ufimsk. Mat. Zh., 13:4 (2021), 126–133; Ufa Math. J., 13:4 (2021), 123–130
Citation in format AMSBIB
\Bibitem{ShaNor21}
\by O.~Sh.~Sharipov, A.~F.~Norjigitov
\paper Law of large numbers for weakly dependent random variables with values in $D\left[0,1\right]$
\jour Ufimsk. Mat. Zh.
\yr 2021
\vol 13
\issue 4
\pages 126--133
\mathnet{http://mi.mathnet.ru/ufa596}
\transl
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 4
\pages 123--130
\crossref{https://doi.org/10.13108/2021-13-4-123}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000734858600002}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85124324630}
Linking options:
  • https://www.mathnet.ru/eng/ufa596
  • https://doi.org/10.13108/2021-13-4-123
  • https://www.mathnet.ru/eng/ufa/v13/i4/p126
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
    Statistics & downloads:
    Abstract page:134
    Russian version PDF:82
    English version PDF:10
    References:20
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024