| Abstract [eng] |
The main objective of this thesis is the extension of limit result for sums of random linear field by using Beveridge–Nelson decomposition. To achieve this goal, we use Beveridge–Nelson decomposition generalization by V. Paulauskas presented in 2010 and D. Marinucci and S. Poghosyan presented in 2001. These works extend results of P.C.B. Phillips and V. Solo for random linear fields, which were formulated for linear processes. The method enables results proved for random fields of innovations apply to random linear fields, generated by these innovations, under some additional assumption on linear filter. After the investigation of Beveridge–Nelson decomposition we came to the conclusion that the best application of the method is for the proof of Central limit theorem. We consider random linear fields generated by different type of innovation. In Chapter 2 we analyze random linear fields generated by martingale difference innovations. Definition of martingale difference in the plane and higher dimension spaces is another important topic analyzed in the thesis, because there exist different ways to define them. We use different definitions of martingale difference presented in the works by D. Tjøstheim, R. Morkvėnas, B. Nahapetian, M. El Machkouri, and prove Central limit theorems for random linear fields with three different types of martingale difference innovations. In the last chapter we consider random linear fields generated by ergodic or mixing (in particular case, independent identically distributed (i.i.d.)) random variables. There we generalize the classical Strong Law of Large Numbers for multi-indexed sums of i.i.d. random variables. These results are easily obtained using ergodic theory. Also we compare the results for SLLN obtained using ergodic theory and with the help of the Beveridge–Nelson decomposition. |
