Local scaling limits of L\'evy driven fractional random fields

Publication date: 1 February 2021

Abstract: We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields

X

on

{m a t h b b 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 (l a m b d a)

and

O (l a m b d a^{g} a m m a)

respectively as

l a m b d a d o w n a r r o w 0

, for some

g a m m a > 0

. We consider two types of increments of

X

: usual increment and rectangular increment, leading to the respective concepts of

g a m m a

-tangent and

g a m m a

-rectangent random fields. We prove that for above

X

both types of local scaling limits exist for any

g a m m a > 0

and undergo a transition, being independent of

g a m m a > g a m m a_{0}

and

g a m m a < g a m m a_{0}

, for some

g a m m a_{0} > 0

; moreover, the "unbalanced" scaling limits (

g a m m a e g a m m a_{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 Pilipauskait.e and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on

{m a t h b b Z}^{2}

and Benassi et al. (2004) on

1

-tangent limits of isotropic fractional L'evy random fields.

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