Remarks on the SLLN for linear random fields

Abstract

We consider random linear fields on Z d generated by ergodic or mixing (in particular case, independent identically distributed (i.i.d.)) random variables. Our main results generalize classical Strong Law of Large Numbers (SLLN) 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.MSC 2000 subject classifications. Primary: 60F10, 60F15, 60G60.

A Simple Approach in Limit Theorems for Linear Random Processes and Fields with Continuous Time

“…Proof of (10). The proof is similar to the proof of Proposition 2: we write Y t = Y (2) t , applying Proposition 1, we have…”

“…Concerning SLLN one can note that application of (6) is almost useless, since usually to get SLLN for the process ε(t), we assume that it is mean zero ergodic and stationary, but then the same properties have the process X(t) and SLLN for it holds (see [2], where SLLN is considered for discrete random fields). We provide only three examples of application of decompositions formulated in the previous subsection.…”

“…It is important that this decomposition be applied to an arbitrary sequence {ε i } (even stationarity is not needed; the only requirement is that X(t) and a new linear processε t are correctly defined). From (2) it follows that…”

“…Taking b k = c k , a k = ε t−k , from (4) we get (2). Although in [9] it is said that identity (2) in the context of time series was known and used before the paper [3], the abbreviation BN decomposition was used (perhaps for the first time) in [9].…”

“…Although in [9] it is said that identity (2) in the context of time series was known and used before the paper [3], the abbreviation BN decomposition was used (perhaps for the first time) in [9]. Now this term Beveridge-Nelson decomposition is so widely used in the econometric literature Downloaded 12/13/14 to 216.165.95.79.…”