h_kr {rlfsm}

Function h_kr

Description

Function h_{k,r}: R \to R is given by

h_{k,r} (x) = \sum_{j=0}^k (-1)^j {{k}\choose{j}} (x-rj)_+^{H-1/\alpha}, \ \ \ x\in R

Usage

h_kr(k, r, x, H, alpha, l = 0)

Arguments

k

order of the increment, a natural number

r

difference step, a natural number

x

real number

H

Hurst parameter

alpha

self-similarity parameter of alpha stable random motion.

l

a value by which we shift x. Is used for computing function f_.+l and is passed to integrate function.

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.

Examples

#### Plot h_kr ####
s<-seq(0,10, by=0.01)
h_val<-sapply(s,h_kr, k=5, r=1, H=0.3, alpha=1)
plot(s,h_val)