Stable Distributions

1.3 Definitions and basic characteristics

slower for smaller data sets. Moreover, the FFT based approach is less uni-
versai - it is efficient only for large α’s and only for pdf calculations. When
computing the cdf the density must be numerically integrated. In contrast, in
the direct integration method Zolotarev’s (1986) formulas either for the density
or the distribution function are numerically integrated.

Set ζ = -β tan ^π2α. Then the density f (x ; α, β) of a standard α-stable random
variable in representation S0, i.e. X ~ S^(1 ,β, 0), can be expressed as (note,
that Zolotarev (1986, Section 2.2) used yet another parametrization):

• when α 6= 1 and x > ζ :

f ( x ; α,β )= ^α∏x^- -/ ² V ( θ ; α,β )exp © - ( x - ζ ) ^α-1 V ( θ ; α,β )} dθ,

^-ξ                                           (1.5)

• when α 6= 1 and x = ζ :

f(x; α, β) =

^r(^{1 + 1 )cos( ξ} )
∏ (1 + ζ² ) 2^α

• when α 6= 1 and x < ζ :

f(x; α, β) = f(-x; α, -β),

• when α = 1:

f 2βe^-πx R-22 V(θ;1 ,β)exp {→-^πx V(θ;1 ,β)} dθ,
f ( χ ;1 ,β ) =

I 1

π(1+x2) ^,

β 6= 0,

β = 0,

where

ʃ¹ arctan(-ζ ), α = 1,

[∏,              α =1,

and

(cos αξ) α-ι ( . ^cos^θ^_fj. ʌ ^{α-1 cos}{^αξ+(α-1)^θ}

^I                  sin α(ξ+θ)              cosθ

V ( θ ; α,β ) =

[ π ( ^π+^βθ ´ exp n¹ (^π+^βθ )^taⁿ θ O^,

α 6= 1,

α = 1, β 6= 0.