The Almost Sure Central Limit Theorem

The almost sure central limit theorem is a pop topic of the probability research in recent years,because it has many actual applications in the random analogue.Recently,there are a lot of works on the almost sure central limit theorem and many important results have been obtained.Let{Xk, k≥1} be a real random variables sequence on probability space (Ω, F, P), n≥1 . Define the convert-line process ,here we suppose that sn(0,ω) = 0.The almost sure central limit theorem problems mainly focus on conditions, on whichandhold.Hereδx occured in (1)is a single-point measure over R,δx occured in (2)is a single-point measure on continuous function space C[0,1], N(0, 1)is a standard normal distribution , and W is the standard Brown movement on [0,1].Let{Xk,k≥1} be i.i.d. random variables sequence, with EX1 = 0,EX₁² = 1, defined on a probability space (Ω, F, P), Put In 1990,Lacey and Philipp proved that (2) hold . Several authors have observed that for i.i.d. sequences , the finite second moment condition is necessary for the ASCLT .It is easy to see that ASCLT problems for i.i.d. sequence are almost solved. It is obvious that (1) is equal to that for any bounded continuous functions f,Where , G denotes standard normal distribution .In 1998, I.Ibragimtv and M.Lifshits proved that for some unbounded functions (3) is still hold with some extra conditions, and I.Berkes,E.Csaki and L.Horvath gave some conditions on which (3)is holds.The almost sure central limit theorem problems for weakly dependent random variables has became one of the most important and popular orientations of the study of ASCLT problems . In 1995 ,Magda Peligrad and Qi-Man Shao gave ASCLT for associated sequence ,strongly mixing andρ-mixing sequence under the same conditions that assure that the central limit theorem hold . In 2004 ,Zhi-Shan Dong and Xiao-Yun Yang proved that NA random variables with stationary distribution satisfy (2) on condition that the second order moment exists.In 2004 ,B.Bercu established new almost sure asymptotic properties for martingales transforms. It enables us to deduce the convergence of moments in the almost sure limit theorem for martingales. Sveral statistical applications on the asymptotic behavior of stochastic regression models were also provided.