Evaluates the pdf for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.
dCTS(x, alpha, c = 1, ell = 1, mu = 0)
Vector of points.
Number in [0,2)
Parameter c >0
Parameter ell>0
Location parameter, any real number
Michael Grabchak and Lijuan Cao
The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*delta_ell(dx) + c*delta_(-ell)(dx), where delta is the delta function. The Levy measure is M(dx) = c*ell^(alpha) *e^(-x/ell)*x^(-1-alpha) dx. The characteristic function is, for alpha not equal 0,1:
f(t) = exp( 2*c*gamma(-alpha)*(1+ell^2 t^2)^(alpha/2)*(cos(alpha*atan(ell*t))-1)) *e^(i*t*mu),
for alpha = 1 it is
f(t) = (1+ell^2 t^2)^c*exp(-2*c*ell*t*atan(ell*t)) *e^(i*t*mu),
and for alpha=0 it is
f(t) = (1+t^2 ell^2)^(-c) *e^(i*t*mu).
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.