Limit theorems for linear random fields with tapered innovations. II: The stable case

Abstract

In the paper, we consider the limit behavior of partial-sum random field (r.f.) $\left.{S}_n\left({t}_1,{t}_2;\right)X\left(b\left(\mathbf{n}\right)\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}\left(b\left(\mathbf{n}\right)\right),$ where $\left\{{X}_{k,l}\left(b\left(\mathbf{n}\right)\right)={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_{i,j}{\upxi}_{k-i,l-j}\left(b\left(\mathbf{n}\right)\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1,$ is a family (indexed by n = (n₁, n₂), n_i ≥ 1) of linear r.f.s with filter c_i,j = a_ib_j and innovations ξ_k,l(b(n)) having heavy-tailed tapered distributions with tapering parameter b(n) growing to infinity as n → ∞. In [V. Paulauskas, Limit theorems for linear random fields with tapered innovations. I: The Gaussian case, Lith. Math. J., 61(2):261–273, 2021], we considered the so-called hard tapering as b(n) grows relatively slowly and the limit r.f.s for appropriately normalized S_n(t₁, t₂;X(b(n))) are Gaussian. In this paper, we consider the case of soft tapering where b(n) grows more rapidly in comparison with the case of hard tapering and stable limit r.f.s.We consider cases where the sequences {a_i} and {b_j} are long-range, short-range, and negatively dependent.