Asymptotics for linear random fields

On Beveridge–Nelson decomposition and limit theorems for linear random fields

“…If in [5] IP (the convergence to a Wiener process) was proved under the natural second-moment condition, all our attempts to prove (we tried direct and indirect applications of BND) an analogous result for fields failed; all calculations showed that the existence of moments of order > 4 is needed. The same result is in the above-cited paper [6], where IP for linear random fields (for an arbitrary d) was proved assuming the existence of moments of innovations of order q > 2d. Thus, the question whether the IP for linear random fields can be proved using BND and under the second-moment condition remains open.…”

“…Later we found that such a decomposition was obtained in a recent paper [6] by Marinucci and Poghosyan; on the other hand, working on BND analogue for fields without knowing the results of [6] had some advantage: we proved some new relations (that were absent in [6]) between the initial coefficients of a linear field and the coefficients in the decomposition.…”

Remarks on the SLLN for linear random fields

“…random variables and the problem to show that R n (in some sense) converges to zero. In Paulauskas (2009) the BND for linear random fields was applied to prove CLT and SLLN, earlier in Marinucci and Poghosyan (2001) BND was used to prove IP. In Paulauskas (2009) it was stressed that in the case d = 2 and sets D n = [1, n] 2 ∩ Z 2 this approach leads to very simple proofs, but at the same time moment conditions for innovations ε t and conditions for coefficients ϕ k are not optimal (this is the price which we pay for simplicity of proofs).…”

“…The main message of our note is that the application of the ergodic theory to prove SLLN for linear fields gives much more general and stronger results comparing with ones obtained by using BND and even the proofs, based on application of ergodic theorems, are very simple. Since the application of BND for IP also faces some difficulties (see Paulauskas (2009) and Marinucci and Poghosyan (2001)), it seems that the most successful application of BND is for the CLT in Paulauskas (2009). Before formulation of our results we introduce some notions from the ergodic theory.…”

Long memory random fields

“…Dans un cadre plus restreint (le champ X est supposé être du type accroissement d'une martingale), Basu et Dorea (1979) montrent le même type de résultat en n'exigeant que 1.3 Revue de l'asymptotique des sommes partielles d'un champ aléatoire 9 des moments d'ordre 2 finis pour X. Pour les champs associés, en supposant l'existence de moments d'ordre strictement supérieur à 2, Bulinski et Keane (1996) et Marinucci et Poghosyan (2001) montrent le même type de convergence pour les sommes partielles indexées par des quadrants.…”