Chebyshevskii Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






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


Chebyshevskii Sbornik, 2014, Volume 15, Issue 3, Pages 86–99 (Mi cheb353)  

On a functional limit theorem for additive functions

Kh. Kh. Usmanov

Volzhsk Branch of Moscow Power Engineering Institute
References:
Abstract: By means of additive arithmetic functions on a sequence of the shifted prime numbers the processes with realizations from a space of functions without ruptures of the second sort are based. In this space with a topology of Skorokhod and $\sigma$-algebra of the borelean multitudes a sequence of the measures corresponding to constructed arithmetic processes is entered. Exactly, the relative frequency of prime numbers is accepted to a measure of the borelean multitudes. These numbers don't surpass natural number of $n$ to which there a correspond realization of the constructed processes getting to this multitude. Necessary and sufficient conditions of weak convergence of sequence of the entered measures to the measure corresponding to a process are found. Thus process with the independent increments, which distributions are not expressed, is limited. Necessary and sufficient conditions represent two limit ratios the first of which is an infinite of a sequence of the set sums. The proofs of need of performance of this ratio for weak convergence of sequence of measures are the main part of all proofs of the theorem. This proof is carried out by consideration of distributions of increments of arithmetic processes on the intervals close to a unit and a transition to characteristic functions, corresponding to these distributions. Further, using an independence of increments of a limit process and a weak compactness of a sequence of measures (taken from Yu. Prokhorov's known theorem of weak convergence of probability measures), by an asymptotic formula for average values of multiplicative functions on sequence of the shifted prime numbers of N. Timofeev, we receive the first condition of the theorem. At the proof of sufficiency of both conditions for weak convergence of sequence of measures characteristic functions are applied again. It allows, in a particular, to use early the limit theorems received by the author in functional spaces for additive functions on “rare” multitudes. The sequence $\{p+1\}$ is included in a class of the sequences considered in these theorems. However, in them the condition similar to the first condition considered here, isn't necessary, but is sufficient. It allows, applying the specified theorems to a considered case to receive a weak convergence of sequence of measures. A representation for a characteristic function of a limit process is also received.
Bibliography: 16 titles.
Keywords: additive function, characteristic function, stochastic process, measure.
Received: 27.06.2014
Document Type: Article
UDC: 511.37
Language: Russian
Citation: Kh. Kh. Usmanov, “On a functional limit theorem for additive functions”, Chebyshevskii Sb., 15:3 (2014), 86–99
Citation in format AMSBIB
\Bibitem{Usm14}
\by Kh.~Kh.~Usmanov
\paper On a functional limit theorem for additive functions
\jour Chebyshevskii Sb.
\yr 2014
\vol 15
\issue 3
\pages 86--99
\mathnet{http://mi.mathnet.ru/cheb353}
Linking options:
  • https://www.mathnet.ru/eng/cheb353
  • https://www.mathnet.ru/eng/cheb/v15/i3/p86
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Statistics & downloads:
    Abstract page:174
    Full-text PDF :104
    References:48
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024