Invariance principle for martingale-difference random fields

A new condition for the invariance principle for stationary random fields

“…Remark 5.4. Our result is more general than Basu and Dorea (1979), Nahapetian (1995) and Poghosyan and Roelly (1998) in the following sense. Let be {ǫ i,j } (i,j)∈Z 2 be i.i.d.…”

“…It is easy to see, in both cases above, their assumptions are stronger, in the sense that they imply that {M i,j } (i,j)∈N 2 is an orthomartingale, with the natural filtration {F i,j } (i,j)∈N 2 (1.4). On the other hand, however, the results in ( Basu and Dorea (1979); Poghosyan and Roelly (1998)) only assume that {ǫ i,j } (i,j)∈Z 2 is a stationary random field, which is weaker than our assumption.…”

An invariance principle for stationary random fields under Hannan’s condition

“…See for example Berkes and Morrow [2], Bolthausen [3], Goldie and Morrow [13], Bradley [4] for results under mixing conditions, Basu and Dorea [1], Morkvėnas [20], Nahapetian [21], Poghosyan and Roelly [23] for results on multiparameter martingales, and Dedecker [7,8], El Machkouri et al [12], Wang and Woodroofe [25] for results on random fields satisfying projective-type assumptions. In particular, projective-type assumptions have been significantly developed for invariance principles for stationary sequences (d = 1).…”

Invariance principle via orthomartingale approximation

“…centered random field with finite variance, which generalized Donsker's one dimensional result [Don51]. Wichura's result was extended to a class of stationary ergodic martingale differences random fields [BD79,PR98], and Dedecker found a projective condition [Ded01]. Wang and Woodroofe [WW13] attempted to extend the Maxwell and Woodroofe condition [MW00] but found a weaker condition, which was improved by Volný and Wang [VW14].…”