Optimal averaging procedures in almost sure central limit theory

Hörmann, Siegfried (author)

Let X1,X2, . . . be i.i.d. random variables with EX1 = 0,EX2 1 = 1, Sn = X1 +.. .+Xn and let (dk) be a positive numerical sequence. We investigate the a.s. convergence of the averages 1 DN N Xk=1 dkI{Sk/¡Ìk ¡Ü x} ,where DN = PN k=1 dk. In the case of dk = 1/k we have logarithmic means and by the almost sure central limit theorem the above averages converge a.s. (x), the standard normal distribution function. It is also known that the analogous convergence relation fails for dk = 1 (ordinary averages). In this paper we give a fairly complete solution of the problem for which weight sequences the above convergence relation and its refinements hold.

• Language: English
• Work type: Not categorized
• Typology: 1.01 - Original Scientific Article
• Organization: FDV - Faculty of Social Sciences
• Year: 2005
• Number of pages: Str. 271-282
• Numbering: Vol. 2, no. 2
• PID: 20.500.12556/RUL-22457
• UDC: 311
• ISSN on article: 1854-0023
• COBISS.SI-ID: 24318301
• Publication date in RUL: 11.07.2014

Record is a part of a journal

• Title: Advances in methodology and statistics
• Shortened title: Metodol. zv.
• Publisher: Fakulteta za družbene vede
• ISSN: 1854-0023
• COBISS.SI-ID: 215795712