Stable Distributions

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 :

^si^gn(^{1 α} ) 12 exp © - ( x - ζ ) ^α-1 V ( θ ; α,β )} dθ,
^π      -ξ

• 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})^,

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 ∏.

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