Local scaling limits of Lévy driven fractional random fields

Pilipauskaitė, Vytautė; Сургайлис, Донатас

doi:10.3150/21-bej1439

Cited by 2 publications

(6 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…i for all c > 0, d < 0, and some c 1 ∈ R. For ν = 2, Ref. [25] Equation (3.7) has recently derived the analytic form in (32) of the kernel from the absolute square of its Fourier transform:…”

Section: Invertibility and Properties Of Fractional Operatorsmentioning

confidence: 99%

“…which is the implicit definition of this kernel in [22]. Similarly to derivations in [25], for ν ≥ 2, Equations (3.944.5-6) in the table of integrals [27] give…”

Section: Invertibility and Properties Of Fractional Operatorsmentioning

confidence: 99%

“…We expect that this study can be extended in several directions, including anisotropic scaling, infinite variance random fields, and fractional operators in R ν (see [20][21][22][23][24][25] for discussion and the properties of fractional random fields with the continuous argument t ∈ R ν ).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractional Operators and Fractionally Integrated Random Fields on Zν

Pilipauskaitė,

Surgailis

2024

Fractal Fract

View full text Add to dashboard Cite

We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=ε 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.

show abstract

Section: Invertibility and Properties Of Fractional Operatorsmentioning

confidence: 99%

“…which is the implicit definition of this kernel in [22]. Similarly to derivations in [25], for ν ≥ 2, Equations (3.944.5-6) in the table of integrals [27] give…”

Section: Invertibility and Properties Of Fractional Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Fractional Operators and Fractionally Integrated Random Fields on Zν

Pilipauskaitė,

Surgailis

2024

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…We expect that this study can be extended in several directions, including anisotropic scaling, infinite variance random fields, and fractional operators in R ν . See [1,6,16,20,21,24] for discussion and properties of fractional random fields with continuous argument t ∈ R ν .…”

Section: Of 20mentioning

confidence: 99%

“…which is implicit definition of this kernel in [16]. Similarly to derivations in [24], for ν ≥ 2, table of integrals [15] [3.944.5-6] gives…”

Section: Invertibility and Properties Of Fractional Operatorsmentioning

confidence: 99%

Fractional Operators and Fractionally Integrated Random Fields on Z

Surgailis,

Pilipauskaitė

2024

Preprint

View full text Add to dashboard Cite

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.

show abstract

Local scaling limits of Lévy driven fractional random fields

Cited by 2 publications

References 36 publications

Fractional Operators and Fractionally Integrated Random Fields on Zν

Fractional Operators and Fractionally Integrated Random Fields on Zν

Fractional Operators and Fractionally Integrated Random Fields on Z

Contact Info

Product

Resources

About