Scaling Limits of Linear Random Fields on $${\mathbb {Z}}^2$$ with General Dependence Axis

Pilipauskaitė, Vytautė; Сургайлис, Донатас

doi:10.1007/978-3-030-60754-8_28

Cited by 5 publications

(11 citation statements)

References 15 publications

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“…Related trichotomy of the scaling behavior was reported in large-scale anisotropic scaling for several classes of long-range dependent (LRD) planar RF models, with rectangular increment replaced by a sum or integral of the values on large rectangle with sides increasing at different rates λ and λ γ as λ → ∞ for any given γ > 0; see [33,34,29,30,31,40]. In the above works, this trichotomy was termed the scaling transition, with V ± the unbalanced and V γ 0 the well-balanced scaling limits.…”

Section: Introductionmentioning

confidence: 79%

“…0 are very usual and extremely singular objects and their appearance in limit theorems is surprising [40]. We note the FBS with (H 1 , H 2 ) = (0, 1/2) or (1/2, 0) as anisotropic partial sums limits of linear RFs on Z 2 with negative dependence and edge effects were obtained in [40]; in the case of LRD RFs on Z 2 unbalanced Gaussian limits were proved to be FBS with at least one of H i , i = 1, 2, equal 1/2 or 1 [30,31].…”

Section: Introductionmentioning

confidence: 84%

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Local scaling limits of Lévy driven fractional random fields

Pilipauskaitė¹,

Сургайлис²

2021

Preprint

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We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields X on R 2 written as stochastic integral with respect to infinitely divisible random measure.The scaling procedure involves increments of X over points the distance between which in the horizontal and vertical directions shrinks as O(λ) and O(λ γ ) respectively as λ ↓ 0, for some γ > 0. We consider two types of increments of X: usual increment and rectangular increment, leading to the respective concepts of γ-tangent and γ-rectangent RF. We prove that for above X both types of local scaling limits exist for any γ > 0 and undergo a transition, being independent of γ > γ 0 and γ < γ 0 , for some γ 0 > 0; moreover, the 'unbalanced' scaling limits (γ = γ 0 ) are (H 1 , H 2 )-multi self-similar with one of H i , i = 1, 2, equal to 0 or 1. The paper extends Pilipauskaitė and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on Z 2 and Benassi et al. ( 2004) on 1-tangent limits of isotropic fractional Lévy random fields.

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 84%

Local scaling limits of Lévy driven fractional random fields

Pilipauskaitė¹,

Сургайлис²

2021

Preprint

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“…Following [17], we may interpret the number M of independent sources growing with T for some γ > 0 in (1.1)-(1.2) as the connection rate, and refer to the cases γ > γ 0 and γ < γ 0 as fast and slow growth for the connection rate, respectively. Related scaling trichotomy (termed scaling transition) was observed for a large class of planar RFs with long-range dependence (LRD), see [31,30,24,25,34,26]. In these works, A λ,γ (x, y) correspond to a sum or integral of values of a stationary RF (indexed by Z 2 or R 2 ) over large rectangle (0, λx] × (0, λ γ y] whose sides increase as O(λ) and O(λ γ ), for a given γ > 0.…”

Section: Introductionmentioning

confidence: 91%

“…More For 0 < t < x, by the strong LLN, λ −1 N (λ(x − t)) → µ −1 (x − t) P 0 -a.s., moreover, by the elementary renewal theorem, λ −1 E 0 N (λ(x − t)) → µ −1 (x − t). Accordingly, the limit of Ψλ (t [7, (28), (26)]. Finally, P(Z > λt) ∼ c Z (λt) −α , λ → ∞, for t > 0, whereas P(Z > λt) ≤ C(λt) −α for t > ǫ λ since λǫ λ → ∞.…”

Section: The Telecom Process and Large Deviationsmentioning

confidence: 99%

“…of (6.16), use the Chebyshev inequality and the fact that E 0 (N (λ) − λ) 2 ≤ Cλ 3−α , λ ≥ 1, see e.g. [7, (28), (26)]. Then λ −1 ∞ Kλ Fλ (u)du ≤ Cλ 2−α ∞ Kλ u −2 du ≤ CK −1 λ 1−α and hence the corresponding term in (6.16) can be made arbitrarily small uniformly in λ ≥ 1 by choosing K > 1 sufficiently large, proving (6.15) for i = 2.…”

Section: The Telecom Process and Large Deviationsmentioning

confidence: 99%

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Aggregation of network traffic and anisotropic scaling of random fields

Leipus¹,

Pilipauskaitė²,

Сургайлис³

2021

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We discuss joint spatial-temporal scaling limits of sums A λ,γ (indexed by (x, y) ∈ R 2 + ) of large number O(λ γ ) of independent copies of integrated input process X = {X(t), t ≥ 0} at time scale λ, for any given γ > 0. We consider two classes of inputs X: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that normalized random fields A λ,γ tend to an α-stable Lévy sheet (1 < α < 2) if γ < γ 0 , and to a fractional Brownian sheet if γ > γ 0 , for some γ 0 > 0. We also prove an 'intermediate' limit for γ = γ 0 . Our results extend previous work [17,7] and other papers to more general and new input processes.

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Fractional Operators and Fractionally Integrated Random Fields on Z

Surgailis,

Pilipauskaitė

2024

Preprint

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We consider fractional integral operators $(I-T)^d, d \in (-1,1)$ acting on functions $g: \mathbb{Z}^{\nu} \to \mathbb{R}, \nu \ge 1 $, where $T $ is the transition operator of a random walk on $\mathbb{Z}^{\nu}$. We obtain sufficient and necessary conditions for the existence, invertibility and square summability of kernels $\tau (\mathbf{s}; d), \mathmb{s} \in \mathbb{Z}^{\nu}$ of $(I-T)^d $. Asymptotic behavior of $\tau (\mathbf{s}; d)$ as $|\mathbf{s}| \to \infty$ is identified following local limit theorem for random walk. A class of fractionally integrated random fields $X$ on $\mathbb{Z}^{\nu}$ solving the difference equation $(I-T)^d X = \varepsilon$ with white noise on the right-hand side is discussed, and their scaling limits. Several examples including fractional lattice Laplace and heat operators are studied in detail.

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Scaling Limits of Linear Random Fields on $${\mathbb {Z}}^2$$ with General Dependence Axis

Cited by 5 publications

References 15 publications

Local scaling limits of Lévy driven fractional random fields

Local scaling limits of Lévy driven fractional random fields

Aggregation of network traffic and anisotropic scaling of random fields

Fractional Operators and Fractionally Integrated Random Fields on Z

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