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74 3. Distributions exponentielles étirées contre distributions régulièrement variables
nent c = 0.3 and c = 0.7. Thus, the three first samples are the iid counterparts of the later ones. The
sample with regularly varying iid distributions converges to the Fréchet’s maximum domain of attraction
with ξ = 1/3 = 0.33, while the iid Stretched-Exponential distribution converges to Gumbel’s maximum domain
of attraction with ξ = 0. We now study how well can one distinguish between these two distributions
belonging to two different maximum domains of attraction.
For the stochastic processes with long memory, we use a simple stochastic volatility model. First, we
construct a Gaussian process {Xt}t≥1 with correlation function
C(t) =
1
(1+|t|) α
if |t| ≤ T ,
0 if |t| > T .
It should be noted that, in order for the stochastic process to be well-defined, the correlation function must
satisfy a positivity condition. More precisely, the spectral density (the Fourier transform of the correlation
function) must remain positive. This condition imposes that the duration of the memory T be larger than a
constant depending on α.
The next step consists in building the process {Ut}t≥1, defined by
(11)
Ut = Φ(Xt) , (12)
where Φ(·) is the Gaussian distribution function. The process {Ut}t≥1 exhibits exactly the same long range
dependence as the process {Xt}t≥1. This is ensured by the property of invariance of the copula under
strickly increasing change of variables. Let us recall that a copula is the mathematical embodiement of the
dependence structure between different random variables (Joe 1997, Nelsen 1998). The process {Ut}t≥1
thus possesses a Gaussian copula dependence structure with long memory and uniform marginals 5 .
In the last step, we define the volatility process
σt = σ0 ·U −1/b
t , (13)
which ensures that the stationary distribution of the volatility is a Pareto distribution with tail index b. Such
a distribution of the volatility is not realistic in the bulk which is found to be approximately a lognormal
distribution for not too large volatilities (Sornette et al. 2000), but is in agreement with the hypothesis of
an asymptotic regularly varying distribution. A change of variable more complicated than (13) can provide
a more realistic behavior of the volatility on the entire range of the distribution but our main goal is not to
provide a realistic stochastic volatility model but only to exhibit a long memory process with well-defined
prescribed marginals in order to test the influence of a long range dependence structure.
The return process is then given by
rt = σt · εt , (14)
where the εt are Gaussian random variables independent from σt. The construction (14) ensures the decorrelation
of the returns at every time lag. The stationary distribution of rt admits the density
p(r) = 2 b−1
b − 1 r2 b
2 · Γ , , (15)
2 2 rb+1
b−1 r2
which is regularly varying at infinity since Γ 2 , 2 goes to Γ
b−1
2 . This completes the construction and
characterization of our long memory process with regularly varying stationary distribution.
5 Of course, one can make the correlation as small as one wants under an adequate choice of a strickly increasing transformation
but this does not change the fact that the dependence remains unchanged. This is another illustration of the fact that the correlation
is not always a good and adapted measure of dependence (Malevergne and Sornette 2002).
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