1 Stable distributions
The distribution F (x; α, β) of a standard α-stable random variable in represen-
tation S0 can be expressed as:
• when α 6= 1 and x > ζ :
F (x; α, β) = c1 (α, β) +
sign(1 α ) 12 exp © - ( x - ζ ) α-1 V ( θ ; α,β )} dθ,
π -ξ
where
c 1( α,β )= 1П(П $’
α< 1,
α> 1,
• when α 6= 1 and x = ζ :
F(χ; α,β) = 1 (П
π2
• when α 6= 1 and x < ζ :
F(x; α, β) = 1 - F (-x; α, -β),
• when α = 1:
∏ Λ r∙ — — π X , . ʌl
∏ L2 — exp {-e-2βV(θ;1 ,β)} dθ, β> 0,
F( x ;1 ,β )=S 1 + 1 arctan x, β = 0,
2π
. 1 - F(χ, 1, -β),
β<0.
Formula (1.5) requires numerical integration of the function g(∙) exp{-g(∙)},
where g(θ; x,α,β) = (x - ζ)α-1 V(θ; α, β). The integrand is 0 at -ξ, increases
monotonically to a maximum of 1 at point θ* for which g(θ* ; x,α,β) = 1,
and then decreases monotonically to 0 at ∏ (Nolan, 1997). However, in some
cases the integrand becomes very peaked and numerical algorithms can miss
the spike and underestimate the integral. To avoid this problem we need to
find the argument θ* of the peak numerically and compute the integral as a
sum of two integrals: one from -ξ to θ* and the other from θ* to ∏.