m_pk {rlfsm}

m(-p,k)

Description

defined as m_{p,k} := E[|\Delta_{k,k} X|^p] for positive powers. When p is negative (-p is positive) the equality does not hold.

Usage

m_pk(k, p, alpha, H, sigma)

Arguments

k

increment order

p

a positive number

alpha

self-similarity parameter of alpha stable random motion.

H

Hurst parameter

sigma

Scale parameter of lfsm

Details

The following identity is used for computations:

m_{-p,k} = \frac{(\sigma \|h_k\|_{\alpha})^{-p}}{a_{-p}} \int_{\R} \exp(-|y|^{\alpha}) |y|^{-1+p} dy = \frac{2(\sigma \|h_k\|_{\alpha})^{-p}}{\alpha a_{-p}} \Gamma(p/\alpha)

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.