Limit Theorems for Linear Random Fields with Tapered Innovations. I: The Gaussian case

Abstract

We consider the limit behavior of partial-sum random field $$ {S}_{\mathrm{n}}\left({t}_1,{t}_2;X\left({b}_{\mathrm{n}}\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}^{\left(\mathrm{n}\right)} $$ , where $$ \left\{{X}_{k,l}^{\left(\mathrm{n}\right)}={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_i{,}_j{\upxi}_{k-i,l-j}\left({b}_{\mathrm{n}}\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1 $$ , is a family (indexed by n = (n1, n2), ni ≥ 1) of linear random fields with filter ci,j = aibj and innovations ξk,l(bn) having heavy-tailed tapered distributions with tapering parameter bn growing to infinity as n → ∞. We consider the so-called hard tapering as bn grows relatively slowly and the limit random fields for appropriately normalized Sn(t1, t2;X(bn)) are Gaussian random fields. We consider all cases where sequences {ai} and {bj} are long-range, short-range, and negative dependent. In the second part (as a separate paper), we will consider the case of soft tapering, where bn grows more rapidly, and limit random fields are stable.