Optimal averaging procedures in almost sure central limit theory - Hörmann, Siegfried (Metodološki zvezki, 2005)
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/!Ik !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.
Keywords: Statistical methods; Statistične metode; matematična statistika; stohastični procesi