Stable Distributions

1.4 Simulation of α-stable variables

The complexity of the problem of simulating sequences of α-stable random
variables results from the fact that there are no analytic expressions for the
inverse F^-¹ of the cumulative distribution function. The first breakthrough
was made by Kanter (1975), who gave a direct method for simulating Sα (1, 1, 0)
random variables, for α < 1. It turned out that this method could be easily
adapted to the general case. Chambers, Mallows, and Stuck (1976) were the
first to give the formulas.

The algorithm for constructing a standard stable random variable X ~ Sα(1,β,0),
in representation (1.2), is the following (Weron, 1996):

• generate a random variable V uniformly distributed on ( —^π, ^π ) and an
independent exponential random variable W with mean 1;

• for α 6= 1 compute:

si{ s sin{^α(v + Bα,β)}

^α,β ∙ {cos(V)}¹ /α '

cos^{^V^—^α⁽ V + Bα,β ) ^}

. W .

(1 -α )/α

(1.6)

where

arctan(β tan ^πα )

Bα,β = ---------------2 ,

2 2 παa ¹ / (2^α)

Sα,β = 1 + β2 tan² Ç—) I

• for α = 1 compute:

^πW cos Vλ ɪ
∏ + βv Л

(1.7)

Given the formulas for simulation of a standard α-stable random variable, we
can easily simulate a stable random variable for all admissible values of the
parameters α, σ, β and μ using the following property: if X ~ S_α (1 ,β, 0) then

{σX + μ, α = 1,

(1.8)

σX + 2 βσ ln σ + μ, α = 1,

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