Stable Distribution
A random variable is said to be stable if a linear combination of two
independent copies of a random sample has the same distribution upto
location and scale parameters.
Let 𝑋
1
and 𝑋
2
be independent copies of random variable X. Then X is
said to be stable if for any constant 𝑎 > 0 and 𝑏 > 0 the random variable
𝑎𝑋
1
+ 𝑏𝑋
2
has the same distribution as 𝑐𝑥 + 𝑑 for some constant c ¿ 0
and d.
The distribution is said to be strictly stable if this holds true with d = 0.
Gaussian and Cauchy distributions are example of stable distributions.
Econophysics, 4th Year Levy stochastic process and limit theorem
Prove that Gaussian distribution is stable
Proof:
We know that probability density function of Gaussian random variable
𝑥 with mean zero and s.d. 𝜎 is,
𝑝(𝑥) =
1
√
2𝜋𝜎
𝑒
−𝑥
2
/2 𝜎
2
....(1)
Now, taking the fourier transform of equation (1), we get
𝜙(𝑡) =
∫
∞
−∞
𝑝(𝑥)𝑒
−𝑖𝑡 𝑥
𝑑𝑥
which is characteristics function of random variable x
∴ 𝜙(𝑡) =
∫
∞
−∞
1
√
2𝜋𝜎
𝑒
−𝑥
2
/2 𝜎
2
𝑒
−𝑖𝑡 𝑥
𝑑𝑥
=
1
√
2𝜋𝜎
∫
∞
−∞
𝑒
−𝑥
2
/2 𝜎
2
−𝑖𝑡 𝑥
𝑑𝑥
Econophysics, 4th Year Levy stochastic process and limit theorem
=
1
√
2𝜋𝜎
∫
∞
−∞
𝑒
−
1
2 𝜎
2
(𝑥
2
−𝑖𝑡 𝑥2 𝜎
2
)
𝑑𝑥
=
1
√
2𝜋𝜎
∫
∞
−∞
𝑒
−
1
2 𝜎
2
[𝑥
2
−2.𝑥.𝑖𝑡 𝜎
2
+(𝑖𝑡 𝜎
2
)
2
−(𝑖𝑡 𝜎
2
)
2
]
𝑑𝑥
=
1
√
2𝜋𝜎
∫
∞
−∞
𝑒
−
1
2 𝜎
2
[𝑥+𝑖𝑡 𝜎
2
]
2
𝑒
1
2 𝜎
2
(𝑖𝑡 𝜎
2
)
2
𝑑𝑥
=
1
√
2𝜋𝜎
𝑒
−
𝑡
2
𝜎
4
2 𝜎
2
∫
∞
−∞
𝑒
−
1
2 𝜎
2
[𝑥+𝑖𝑡 𝜎
2
]
2
𝑑𝑥
Put 𝑥 +𝑖𝑡𝜎
2
= 𝑢 =⇒ 𝑑𝑥 = 𝑑𝑢
∴ 𝜙(𝑡) =
1
√
2𝜋𝜎
𝑒
−
𝑡
2
𝜎
2
2
∫
∞
−∞
𝑒
−
1
2 𝜎
2
𝑢
2
𝑑𝑢
=
1
√
2𝜋𝜎
𝑒
−
𝑡
2
𝜎
2
2
𝜋
1/2𝜎
2
∵
∫
∞
−∞
𝑒
−𝑎𝑥
2
𝑑𝑥 =
𝜋
𝑎
Econophysics, 4th Year Levy stochastic process and limit theorem
=
1
√
2𝜋𝜎
𝑒
−𝛾𝑡
2
√
2𝜋𝜎
2
where 𝛾 =
𝜎
2
2
∴ 𝜙(𝑡) = 𝑒
−𝛾𝑡
2
....(2)
According to convolution theorem, the fourier transofrm of convolution
of two function is the product of the fourier transform of the two
functions.
i.e., F( 𝑓 (𝑥) ⊗ 𝑔(𝑥)) = F( 𝑓 (𝑥)) ⊗ F(𝑔(𝑥))
= 𝐹 (𝑡) ⊗ 𝐺(𝑡)
for independent and identically distributed variables, 𝑠
2
= 𝑥
1
+ 𝑥
2
the
probability distribution function 𝑝(𝑠
2
) of sum of 𝑥
1
and 𝑥
2
is given by the
convolution of two probability density function of each random variable
𝑝
2
(𝑠
2
) = 𝑝(𝑥
1
) ⊗ 𝑝(𝑥
2
) ....(3)
Econophysics, 4th Year Levy stochastic process and limit theorem
Taking fourier transform of equation (3)
F[𝑝
2
(𝑠
2
)] = F[𝑝(𝑥
1
) ⊗ 𝑝(𝑥
2
)]
or, 𝜙
2
(𝑡) = F[𝑝(𝑥
1
)] ⊗ F[𝑝(𝑥
2
)]
= 𝜙(𝑡).𝜙(𝑡) ∴ 𝜙
2
(𝑡) = [𝜙(𝑡)]
2
....(4)
𝜙
2
(𝑡) is the characteristics function of 𝑠
2
So, in general, for
𝑠
𝑛
= 𝑥
1
+ 𝑥
2
+ .....𝑥
𝑛
𝑝
𝑛
(𝑠
𝑛
) = 𝑝(𝑥
1
) ⊗ 𝑝(𝑥
2
) ⊗ ..... ⊗ 𝑝(𝑥
𝑛
)
and 𝜙
𝑛
(𝑡) = [𝜙(𝑡)]
𝑛
using (2) and (4) we obtain,
𝜙
2
(𝑡) = (𝑒
−𝛾𝑡
)
2
= 𝑒
−2𝛾𝑡
....(5)
Econophysics, 4th Year Levy stochastic process and limit theorem
Taking the inverse fourier transform of equation (5) we obtain
F[𝜙
2
(𝑡)] = 𝑃
2
(𝑆
2
) =
1
√
2𝜋
∫
∞
−∞
𝜙
2
(𝑡)𝑒
𝑖𝑡 𝑥
𝑑𝑡
or, 𝑃
2
(𝑆
2
) =
1
2𝜋
∫
∞
−∞
𝑒
−2𝛾𝑡
2
𝑒
𝑖𝑡 𝑥
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
−2𝛾𝑡
2
−𝑖𝑡 𝑥
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
−2𝛾 (𝑡
2
−
𝑖𝑡 𝑥
2𝛾
)
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
−2𝛾
h
𝑡
2
−2𝑡
𝑖𝑥
4𝛾
+(
𝑖𝑥
4𝛾
)
2
−(
𝑖𝑥
4𝛾
)
2
i
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
−2𝛾
h
𝑡
2
−
𝑖𝑥
4𝛾
i
2
𝑒
−2𝛾
𝑖
2
𝑥
2
15𝛾
2
𝑑𝑡
=
1
2𝜋
𝑒
−𝑥
2
8𝛾
∫
∞
−∞
𝑒
−2𝛾
h
𝑡
2
−
𝑖𝑥
4𝛾
i
2
𝑑𝑡
Econophysics, 4th Year Levy stochastic process and limit theorem
Put 𝑡 −
𝑖𝑥
4𝛾
= 𝑢 =⇒ 𝑑𝑡 = 𝑑𝑢
𝑝
2
(𝑠
2
) =
1
2𝜋
𝑒
−𝑥
2
8𝛾
∫
∞
−∞
𝑒
−2𝛾𝑢
2
𝑑𝑢 =
1
2𝜋
𝑒
−𝑥
2
8𝛾
𝜋
2𝛾
=
1
√
8𝜋𝛾
𝑒
−𝑥
2
8𝛾
=
1
√
4𝜋𝜎
2
𝑒
−𝑥
2
4 𝜎
2
[∵ 𝛾 =
𝜎
2
2
]
=
1
√
2𝜋
√
2𝜎
2
𝑒
−𝑥
2
2. (2 𝜎
2
)
=
1
√
2𝜋
√
2𝜎
𝑒
−𝑥
2
2. (
√
2 𝜎)
2
let 𝜎
2
=
√
2𝜎, then, 𝑝
2
(𝑠
2
) =
1
√
2𝜋𝜎
2
𝑒
−𝑥
2
2 𝜎
2
2
...(6)
from equation (1) and (6), we see that probability density functions of 𝑥
and 𝑥
1
+ 𝑥
2
are same which proves that the Gaussian distribution is
stable.
Econophysics, 4th Year Levy stochastic process and limit theorem
Show that Lorentzian distribution is stable.
Let us consider a random variable 𝑥 which follows Lorentzian
distribution with probability density function is given by
𝑝(𝑥) =
𝑎
𝜋(𝑥
2
+ 𝑎
2
)
with 𝑥 > 0 ...(1)
where 𝑎 is a parameter.
Now, the fourier transform of (1) is
𝜙(𝑡) = F[𝑝(𝑥)]
=
∫
∞
−∞
𝑝(𝑥)𝑒
𝑖𝑡 𝑥
𝑑𝑥
or, 𝜙(𝑡) =
∫
∞
−∞
𝑎
𝜋(𝑥
2
+ 𝑎
2
)
𝑒
𝑖𝑡 𝑥
𝑑𝑥
= 𝑒
−|𝑡 |𝑎
...(2)
Econophysics, 4th Year Levy stochastic process and limit theorem
Residue = lim
𝑧→𝑖𝑎
(𝑧 −𝑖𝑎)𝑒
𝑖𝑡 𝑧
(𝑧 −𝑖𝑎)(𝑧 + 𝑖𝑎)
=
𝑒
𝑖𝑡 .𝑖𝑎
𝜎𝑎 +𝑖𝑎
=
𝑒
−𝑎𝑡
𝑧𝑖𝑎
But, 𝑡 > 0
∴ 𝑅𝑒𝑠𝑖𝑑𝑢𝑒 =
𝑒
−𝑎|𝑡 |
𝑧𝑖𝑎
∫
∞
−∞
𝑒
𝑖𝑡 𝑧
𝑧
2
+ 𝑎
2
= 2𝜋𝑖
𝑅𝑒𝑠𝑖𝑑𝑢𝑒
= 2𝜋𝑖
𝑒
−𝑎|𝑡 |
𝑧𝑖𝑎
=
𝜋
𝑎
𝑒
−𝑎|𝑡 |
for independent and identically distributed random variables 𝑥
1
and 𝑥
2
distribution is stable.
Econophysics, 4th Year Levy stochastic process and limit theorem
Let 𝑠
2
= 𝑥
1
+ 𝑥
2
The probability density function of the sum of two independent and
identically distributed random variables 𝑥
1
and 𝑥
2
distribution is given
by two convolution of two probability density function of each of them
i.e., 𝑝
2
(𝑠
2
) = 𝑝(𝑥
1
) ⊗ 𝑝(𝑥
2
)
The convolution theorem then implies that the characteristics function
𝜙
2
(𝑡) of 𝑠
2
is given by
F[𝑝
2
(𝑠
2
)] = F[𝑝(𝑥
1
) ⊗ 𝑝(𝑥
2
)]
= F(𝑝(𝑥
1
)) ⊗ F(𝑝(𝑥
2
))
= 𝜙(𝑡) ⊗ 𝜙(𝑡)
∴ 𝜙
2
(𝑡) = [𝜙(𝑡)]
2
....(3)
So, using equation (2) and(3) we get
𝜙
2
(𝑡) = [𝑒
−|𝑡 |𝑎
]
2
= 𝑒
−2|𝑡 |𝑎
....(4)
Econophysics, 4th Year Levy stochastic process and limit theorem
Now taking the inverse fourier transform of (4), we get
𝑝
2
(𝑠
2
) = F[𝜙
2
(𝑡)]
=
1
2𝜋
∫
∞
−∞
𝜙
2
(𝑡)𝑒
−𝑖𝑡 𝑥
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
−2|𝑡 |𝑎
𝑒
−𝑖𝑡 𝑥
𝑑𝑡
=
1
2𝜋
∫
0
−∞
𝑒
2𝑎𝑡 −𝑖𝑡 𝑥
𝑑𝑡 +
∫
∞
0
𝑒
−2𝑡 𝑎−𝑖𝑡 𝑥
𝑑𝑡
=
1
2𝜋
∫
−∞
0
𝑒
2𝑎𝑡 −𝑖𝑡 𝑥
(−𝑑𝑡) +
∫
∞
0
𝑒
−2𝑡 𝑎−𝑖𝑡 𝑥
𝑑𝑡
put t = -u in first integral 𝑝
2
(𝑠
2
)
=
1
2𝜋
∫
−∞
0
𝑒
2𝑎𝑡 −𝑖𝑡 𝑥
(−𝑑𝑡) +
∫
∞
0
𝑒
−(2𝑎+𝑖𝑥)
𝑡𝑑𝑡
Econophysics, 4th Year Levy stochastic process and limit theorem
put −𝑡 = 𝑢 in first integral, then
𝑝
2
(𝑠
2
) =
1
2𝜋
∫
∞
0
𝑒
−2𝑎𝑢+𝑖𝑢𝑥
(𝑑𝑢) +
∫
∞
0
𝑒
−(2𝑎+𝑖𝑥)
𝑡𝑑𝑡
=
1
2𝜋
∫
∞
0
𝑒
−(2𝑎−𝑖 𝑥)𝑢
(𝑑𝑢) +
∫
∞
0
𝑒
−(2𝑎+𝑖𝑥)
𝑡𝑑𝑡
=
1
2𝜋
"
𝑒
−(2𝑎−𝑖 𝑥)𝑢
2𝑎 −𝑖𝑥
∞
0
+
𝑒
−(2𝑎+𝑖𝑥)𝑡
2𝑎 +𝑖𝑥
∞
0
#
=
1
2𝜋
−0 +
𝑒
0
2𝑎 −𝑖𝑥
− 0
𝑒
0
2𝑎 +𝑖𝑥
=
1
2𝜋
1
2𝑎 −𝑖𝑥
+
1
2𝑎 +𝑖𝑥
=
4𝑎
2𝜋
1
4𝑎
2
+ 𝑥
2
∴ 𝑝
2
(𝑠
2
) =
2𝑎
𝜋((2𝑎)
2
+ 𝑥
2
)
....(5)
Equation (1) and (3) shows that the probability density function of 𝑥
and 𝑥
1
+ 𝑥
2
are same. Hence the lorentzian distribution is stable.
Econophysics, 4th Year Levy stochastic process and limit theorem
General Stable distribution
Q. Show that the Levy stable distribution with 𝛼 < 2 has
infinite variance. The Lorentzian distribution and Gaussian
distribution are stable distribution. Their characteristics function is,
𝜙(𝑡) = 𝑒
−𝛾 |𝑡 |
𝛼
....(1)
where, 𝛼 = 1 for Lorentzian and 𝛼 = 2 for Gaussian
The most general form of a characteristics function of a stable process
corresponding to entire class of stable distribution is,
𝑙𝑛[𝜙(𝑡)] =
(
𝑖𝜇𝑡 − 𝛾|𝑡|
𝛼
[1 −𝑖𝛽
𝑡
|𝑡 |
𝑡𝑎𝑛(
𝜋
2
𝛼)] for 𝛼 ≠ 1
𝑖𝜇𝑡 − 𝛾|𝑡|[1 +𝑖𝛽
𝑡
|𝑡 |
2
𝜋
𝑙𝑛|𝑡|] for 𝛼 = 1
....(2)
where 0 < 𝛼 ≤ 2
𝛾 is a positive scale factor 𝜇 is any reak number (population mean)
𝛽 is an assymetry parameter ranging from -1 to +1.
Econophysics, 4th Year Levy stochastic process and limit theorem
In expression (2)
1
If 𝛼 =
1
2
and 𝛽 = 1 the distribution is Levy-simrov.
2
if 𝛼 = 1 and 𝛽 = 0, the distribution is Lorentzian.
3
If 𝛼 = 2 the distribution is Gaussian
Equation (2) is valid for all types of stable distribution. But for
symmetric stable distribution of type (1) we should take 𝜇 = 0 and 𝛽 = 0.
The symmetric stable distribution of index 𝛼 and scale factor 𝛾 is
𝑝(𝑥) =
1
2𝜋
∫
∞
−∞
𝜙(𝑡)𝑒
−𝑖𝑡 𝑥
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
−𝛾 |𝑡 |
𝛼
𝑒
−𝑖𝑡 𝑥
𝑑𝑡
≈
1
𝜋
∫
∞
0
𝑒
−𝛾 |𝑡 |
𝛼
cos 𝑡𝑥𝑑𝑡
∵ 𝑒
−𝑖𝑡 𝑥
= cos 𝑡𝑥 − 𝑖 sin 𝑡𝑥
𝑗
𝑅𝑒(𝑒
𝑖𝑡 𝑥
) = cos 𝑡𝑥
Econophysics, 4th Year Levy stochastic process and limit theorem
for 𝛾 = 1, series expansion of above expression for |𝑥| > 0 results to
𝑝(𝑥) =
−1
𝜋
𝑛
𝑘=1
(−1)
𝑘
𝑘!
√
𝛼𝑘 + 1
|𝑥|
𝑎𝑘+1
sin
𝑘𝜋𝛼
2
+ 𝑅[|𝑥|] ...(3)
where 𝑅[|𝑥|] = 𝑂(|𝑥|
−𝛼(𝑛+1)−1
)
From equation (3) we find that the Asymptotic approximation of a
stable distribution of index 𝛼 is valid for range value of |𝑥| is,
𝑝[|𝑥|] ∼
sin
(𝜋 𝛼)
2
𝜋|𝑥|
1+𝛼
∼ |𝑧|
−(1+𝛼)
The asymptotic behaviour of large values of 𝑥 is a power law behaviour
for this. For this 𝐸 (|𝑥|
𝑛
) i.e., the nth moment diverges when 𝑛 ≥ 𝛼. All
the Levy stable process with 𝛼 < 2 have infinite variance. Thus
non-Gaussian stable stochastic process do not have characteristics length
i.e., the variance for non-Gaussian stochastic process is infinite.
Econophysics, 4th Year Levy stochastic process and limit theorem
Self Similarity
If any system is such that its subunit is analogous or identical to the
overall system’s structure then the system is called a self similar system.
For example, a part of a line segment is itself a line segment and thus a
line segment exhibits self-similarity. By constract, no part of circle is a
circle, and thus a coircle does not exhibit self similarity. Many natural
phenomenon such as cloud and plants are self-similar to some degree.
The distribution which when truncated gives the same structure as the
previous one is called self similar distribution.
In mathematics, self - similar object is exactly or approximately similar
to a part of itself. The distribution which are invariant under stable
scaling of their parameters. Stochastic distribution are self-similar.
Econophysics, 4th Year Levy stochastic process and limit theorem
Rescaling of a non-Gaussian stable distribution toReveal its
self similarity
or How do we rescale non-Gaussian stable distribution to
reveal its self-similarity?
Consider 𝑥 − 1, 𝑥
2
, ...𝑥
𝑛
are independent and identically distributed
random variables following a non-Gaussian stable distribution with
characteristics fnction.
𝜙(𝑡) = 𝑒
−𝛾 |𝑡 |
𝛼
....(1) 0 < 𝛼 < 2
for 𝑆
𝑛
=
𝑛
𝑖=1
𝑥
𝑖
then, 𝜙
𝑛
(𝑡) = 𝑒
𝑛𝛾 |𝑡 |
𝛼
....(2)
where 𝛼 =
(
1 for lorentzian
2 for gaussian
Econophysics, 4th Year Levy stochastic process and limit theorem
Then the probability density function of 𝑆
𝑛
is
𝑃(𝑆
𝑛
) =
1
2𝜋
∫
∞
−∞
𝜙
𝑛
(𝑡)𝑒
𝑖𝑡𝑆
𝑛
𝑑𝑡
=
1
2𝜋
∫
∞
−∞
𝑒
𝑛𝛾 |𝑡 |
𝛼
(cos 𝑡𝑆
𝑛
− sin 𝑡𝑆
𝑛
)𝑑𝑡
for the symmetric stable distribution
𝑃(𝑆
𝑛
) =
1
𝜋
∫
∞
0
𝑒
𝑛𝛾 |𝑡 |
𝛼
(cos 𝑡𝑆
𝑛
− sin 𝑡𝑆
𝑛
)𝑑𝑡
At 𝑆
𝑛
= 0
𝑃(𝑆
𝑛
= 0) =
1
𝜋
∫
∞
0
𝑒
−𝑛𝛾 |𝑡 |
𝛼
(1 − 0)𝑑𝑡
1
𝜋
∫
∞
0
𝑒
𝑛𝛾 |𝑡 |
𝛼
(sin 𝑡𝑆
𝑛
𝑑𝑡 = 0
=
1
𝜋
∫
∞
0
𝑒
−𝑛𝛾 |𝑡 |
𝛼
𝑑𝑡
Econophysics, 4th Year Levy stochastic process and limit theorem
Then, 𝑃(𝑆
𝑛
= 0) =
1
𝜋
∫
∞
0
𝑒
−𝑛𝛾𝑢
𝑑𝑢
𝛼
𝑢
−(1−
1
𝛼
)
𝑃(𝑆
𝑛
) =
1
𝜋𝛼
∫
∞
0
𝑒
−𝑛𝛾𝑢
𝑢
1
𝛼
−1
𝑑𝑢
=
1
𝜋𝛼
Γ(
1
𝛼
)
(𝑛𝛾)
1
1/𝛼
=
1
𝜋𝛼
Γ(
1
𝛼
)
(𝑛𝛾)𝛼
from above 𝑃(𝑆
𝑛
) distribution isproperlyrescaled by defining
˜
𝑝(
˜
𝑆
𝑛
) = 𝑃(𝑆
𝑛
)𝑛
1
𝛼
....(3)
The normalization condition for rescaled distribution is,
∫
∞
0
˜
𝑝(
˜
𝑆
𝑛
)𝑑
˜
𝑆
𝑛
= 1
The normalization of rescaled distribution is assured if
˜
𝑆
𝑛
=
𝑆
𝑛
𝑛
1
𝛼
....(4)
Equation (3) and (4) are the rescaling schemes for non-Gaussian stable
distribution and random variable following that distribution.
Econophysics, 4th Year Levy stochastic process and limit theorem
Infinitely Divisible Random Process
A random process is infinitely divisible if, for every natural number K, it
cam be represented as the sum of K independent and identically
distributed random variables 𝑥
1
. Thedistribution 𝐹 (𝑦) iinfnitely divisible
if and only if the characteristics function 𝜙(𝑞) is, for every natural
number k, the 𝑘
𝑡 ℎ
power of some characteristics function 𝜙
𝑘
(𝑞) in formal
terms
𝜙(𝑞) = [𝜙
𝑘
(𝑞)]
𝑘
The requirement to be divisible random process.
(𝑖)𝜙
𝑘
(0) = 1
(𝑖𝑖)𝜙
𝑘
(𝑞) is continous.
Econophysics, 4th Year Levy stochastic process and limit theorem
Find the characteristics function for poission distribution and
show hat it’s charcteristics fucntion is infinitely divisible
random process.
Let us consider 𝑥 be a discrete random variable following poission
distribution with pmf.
𝑝(𝑥) =
𝑒
−𝜆
.𝜆
𝑥
𝑥!
for x = 0,1, 2, . . n and 𝜆 ≥ 0
Then its characteristics function is,
𝜙(𝑡) = 𝐸 [𝑒
𝑖𝑡 𝑥
]
=
𝑛
𝑥=1
𝑒
𝑖𝑡 𝑥
𝑝(𝑥)
=
𝑥=0
𝑒
𝑖𝑡 𝑥
𝑒
−𝜆
.𝜆
𝑥
𝑥!
= 𝑒
−𝜆
𝑥
(𝑒
𝑖𝑡
𝜆)
𝑥
𝑥!
Econophysics, 4th Year Levy stochastic process and limit theorem
= 𝑒
−𝜆
𝑒
(𝜆𝑒
𝑖𝑡
)
= 𝑒
(−𝜆+𝜆𝑒
𝑖𝑡
)
∴ 𝜙(𝑡) = 𝑒
𝜆(𝑒
𝑖𝑡
−1)
...(1)
This is the characteristics function of poission distribution.
To check infinitely divisibility:
(i) put t = 0 in equation (1)
𝑖.𝑒., 𝜙(0) = 𝑒
𝜆(𝑒
𝑜
−1)
= 𝑒
𝜆(1−1)
= 𝑒
𝑜
or, 𝜙(0) = 1
(ii)
𝜙(𝑡) = [𝑒
𝜆/𝑘 (𝑒
𝑖𝑡
−1)
]
𝑘
= [𝜙
𝑘
(𝑡)]
𝑘
Also 𝜙
𝑘
(0) = 1
As 𝜙(𝑡) satisfies (i) and (ii), it is infinitely divisible random process i.e.,
poission distribution is infinitely divisible (stable/self similar). As finite
divisible random process are stable, poission distribution is stable.
Econophysics, 4th Year Levy stochastic process and limit theorem
Show that normal distribution is infinitely divisible random
process.
Let us consider 𝑥 be independent and identically distributed random
variable following normal distributing with mean 𝜇 and variance 𝜎.
Then the probability density function is
𝑝(𝑥) =
1
√
2𝜋𝜎
𝑒
−
𝑥−𝜇
2 𝜎
2
...(1)
Its characteristics function is given by
𝜙
𝑥
(𝑡) =
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
1
𝜎
√
2𝜋
𝑒
−
𝑥−𝜇
2 𝜎
2
2
𝑑𝑥 ...(2)
Now, putting 𝑧 =
𝑥−𝜇
𝜎
; 𝑑𝑥 =
𝑑𝑥
𝜎
and 𝑥 = 𝑧𝜎 + 𝜇
𝜙
𝑧
(𝑡) =
1
√
2𝜋
∫
∞
−∞
𝑒
𝑖𝑡 (𝜎𝑧+𝜇)
𝑒
−
𝑧
2
2
𝑑𝑧
=
1
√
2𝜋
𝑒
𝑖𝜇𝑡
∫
∞
−∞
𝑒
𝑖𝑡 𝜎𝑧
𝑒
−
𝑧
2
2
𝑑𝑧
Econophysics, 4th Year Levy stochastic process and limit theorem
Again putting 𝑡𝜎 = 𝑘 and 𝛼 = −
1
2
so,
𝜙
𝑧
(𝑡) =
1
√
2𝜋
𝑒
𝑖𝜇𝑡
∫
∞
−∞
𝑒
𝑖𝑡 𝑧
𝑒
−𝛼𝑧
2
𝑑𝑧
=
1
√
2𝜋
𝑒
𝑖𝜇𝑡
∫
∞
−∞
(cos 𝑘𝑧 + 𝑖 sin 𝑘𝑧)𝑒
−𝛼𝑧
2
𝑑𝑧
=
1
√
2𝜋
𝑒
𝑖𝜇𝑡
∫
∞
−∞
cos 𝑘 𝑧.𝑒
−𝛼𝑧
2
𝑑𝑧 +
1
√
2𝜋
𝑒
𝑖𝜇𝑡
∫
∞
−∞
sin 𝑘 𝑧.𝑒
−𝛼𝑧
2
𝑑𝑧
2nd term is zero because integral have an odd function
∴ 𝜙
𝑧
(𝑡) =
1
√
2𝜋
𝑒
𝑖𝜇𝑡
∫
∞
−𝑖𝑛 𝑓 𝑡 𝑦
cos 𝑘 𝑧.𝑒
−𝛼𝑧
2
𝑑𝑧
using standard integral
∫
∞
−∞
cos 𝑘 𝑧.𝑒
−𝛼𝑧
2
𝑑𝑧 =
𝜋
𝛼
𝑒
−𝑘
2
/4𝛼
or, 𝜙
𝑧
(𝑡) =
1
√
2𝜋
𝑒
𝑖𝜇𝑡
𝜋
𝛼
𝑒
−𝑘
2
/4𝛼
...(3)
Econophysics, 4th Year Levy stochastic process and limit theorem
putting back the value of 𝑘 and 𝛼 in Equation (3)
𝜙
𝑧
(𝑡) =
𝑒
𝑖𝜇𝑡
√
2𝜋
√
2𝜋𝑒
−𝑡
2
𝜎
2
/2
= 𝑒
𝑖𝜇𝑡 −
1
2
𝑡
2
𝜎
2
∴ 𝜙
𝑧
(𝑡) = 𝑒
𝑖𝜇𝑡 −
1
2
𝑡
2
𝜎
2
Now to show the stable processes, characteristics fucntion 𝜙(𝑡) should
satisfies two properties.
(𝑖)𝜙(0) = 1
(𝑖𝑖)𝜙(𝑡) = [𝜙
𝑘
(𝑡)]
𝑘
Let check wheather 𝜙(𝑡) satisfies these properties or not
(𝑖)𝜙(0) = 𝑒
[𝑖 𝜇𝑡 −
𝜎
2
2
𝑡
2
]
= 𝑒
[𝑖 𝜇0−
𝜎
2
2
0
2
]
= 𝑒
0
= 1
(𝑖𝑖)𝜙
𝑧
(𝑡) = 𝑒
[𝑖 𝜇
𝑡
𝑘
−
𝜎
2
2
𝑡
2
𝑘
]
𝑘
= [𝜙
𝑘
(𝑡)]
𝑘
This shows that normal distributon is infinitely divisible random process.
Econophysics, 4th Year Levy stochastic process and limit theorem
Illustrate that Gamma distribution is infinitely divisible random process.
Let us consider continous random varible 𝑥 following Gamma
distribution with probability density function.
𝜙
𝑥
=
(
𝑒
−𝑥
𝑥
𝑛−1
Γ (𝑛)
; 𝑛 > 0 ; 0 < 𝑥 < ∞
0 ; otherwise
Now, the characteristics function is,
𝜙
𝑥
(𝑡) = 𝐸 (𝑒
𝑖𝑡 𝑥
) =
∫
∞
0
𝑒
𝑖𝑡 𝑥
𝑝(𝑥)𝑑𝑥 =
∫
∞
0
𝑒
𝑖𝑡 𝑥
𝑒
−𝑥
𝑥
𝑛−1
Γ(𝑛)
𝑑𝑥
=
1
Γ(𝑛)
∫
∞
0
𝑒
𝑖𝑡 𝑥 −𝑥
𝑥
𝑛−1
𝑑𝑥
=
1
Γ(𝑛)
Γ(𝑛)
(1 −𝑖𝑡)
𝑛
∴ 𝜙
𝑥
(𝑡) = (1 −𝑖𝑡)
−𝑛
....(1)
This is the characteristics function of Gamma distribution with
0 < 𝑥 < ∞ and 𝑛 > 0.
Econophysics, 4th Year Levy stochastic process and limit theorem
To check infinitely divisibility:
(i) put 𝑡 = 0 in equation (1)
𝜙
𝑥
(𝑡 = 0) = (1 − 0)
−𝑛
∴ 𝜙
𝑥
(0) = 1
(ii) 𝜙
𝑥
(𝑡) = [(1 −𝑖𝑡)
−𝑛/𝑘
]
𝑘
= [𝜙
𝑘
(𝑡)]
𝑘
with 𝜙
𝑘
(0) = 1
As the characteristics function of Gamma distribution satisfies (i) and
(ii), it is infinitely divisible random process. As infinitely divisible
random process are stable Gamma distribution is also stable
distribution.
Econophysics, 4th Year Levy stochastic process and limit theorem
Q. Show that
𝑝(𝑥) =
(
0 for 𝑥 < −𝑙, 𝑥 > 𝑙
1
2𝑙
for − 𝑙 ≤ 𝑥 ≤ 𝑙
is not an infinitely divisible random process.
Solution
Let us consider a random variable 𝑥 with probability density function.
𝑝(𝑥) =
(
0 for 𝑥 < −𝑙, 𝑥 > 𝑙
1
2𝑙
for − 𝑙 ≤ 𝑥 ≤ 𝑙
Then the characteristics fucntion is
𝜙
𝑥
(𝑡) =
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
𝑝(𝑥)𝑑𝑥
=
∫
𝑙
−𝑙
1
2𝑙
𝑒
𝑖𝑡 𝑥
𝑑𝑥 =
1
2𝑙
𝑒
𝑖𝑡 𝑥
𝑖𝑡
𝑙
−𝑙
Econophysics, 4th Year Levy stochastic process and limit theorem
=
1
2𝑖𝑡𝑥
𝑒
𝑖𝑡𝑙
− 𝑒
−𝑖𝑡𝑙
=
1
𝑡𝑙
𝑒
𝑖𝑡𝑙
− 𝑒
−𝑖𝑡𝑙
2𝑖
=
1
𝑡𝑙
sin 𝑡𝑙
∴ 𝜙
𝑥
(𝑡) =
sin 𝑡𝑙
𝑡𝑙
...(1)
which is the characteristics function of the given distribution
To check infinitely divisiblity:
(i) put 𝑡 = 0 in equation (1)
𝜙
𝑥
(0) = lim
𝑡→0
sin 𝑡𝑙/𝑘
𝑡𝑙
= 1
(ii) 𝜙
𝑥
(𝑡) = [
sin
𝑡𝑙
𝑘
𝑡𝑙
𝑘
]
𝑘
but sin
𝑡𝑙
𝑘
is not contonous for large value of 𝑘. That means roots of
sin
𝑡𝑙
𝑘
𝑡𝑙
𝑘
do not exit. So, the given process is not infinitely divisible.
Econophysics, 4th Year Levy stochastic process and limit theorem
Power law distribution
The distribution of the form |𝑥|
−𝛽
with 𝛽 > 0 is called the power law
distribution. They lack characteris.tics length and most of them are
non-Gaussian stable process moments which are infinite (all moments of
non-Gaussian distribution do not exist)
Physically, for isolated system power law of distributin are meaningless
but for non-isolated system like in socio-economic science, power law
distribution is very useful. They are counter intative they lack
characteristics scale.
Econophysics, 4th Year Levy stochastic process and limit theorem
Power law in infinite system
Describe the role of power laws in infinite systems.
Today, power law distribution are used in the description of open
systems. However, the scaling observed is often limited by finite size
effects or some other limitation instrinsic to the system. A good example
of the fruitful use of power laws and the difficulties related to their use is
provided by critical phenomena. Power law correction functions are
observed in the critical state of an infinite system, but if the system is
finite, the finiteness limits the range within which a power behaviour is
observed. In spite of this limitation, the introduction and the use ofthe
concept of scaling which is related to the power law nature of critical
phenomenon and even when finite systems are considered.
Econophysics, 4th Year Levy stochastic process and limit theorem
Power change statistics
Stable non-Gaussian distribution are the distributions whose all order
moments may not exit. We expect that the price changes in financial
markets are distributed like non-Gaussian but not stable. The limit
theorem for stable distributions are such that the random variable 𝑥
′
𝑖
𝑠 are
(i) Pairwise - independent and
(ii) indentically distributed
For the time series of market, hypothesis (i) an be verified fo time
horizons ranging from a few minutes to several years. However
hypothesis (ii) is not generally verified by emperical observation because
the s.d. of price change is strongly time dependent.
This phenomenon is known in finance as time dependent volatility.That
is why, we should not expect market’s distribution to be stable.
A more appropriate limit theorem isone based only on the assumption
that random variable 𝑥
𝑖
are independent but not necessarilly identically
distributed.
Econophysics, 4th Year Levy stochastic process and limit theorem
St. Petersburg Paradox
St. Peterburg Paradox is related to probability and decision theory in
economics. It is based on a particular (theoretical) Lottery game that
lends to random variable with infinite expected value (i.e. infinite
expected pay (off) but neverless seems to be worth only the expected
value into account predicts a course of action that preumably no actual
person would be willing to take several resolutions are possible. The
paradox takes its name from its resolution by Daniel Bernouli one-time
resisdent of eponymous Russian city, who published his arguements in
the commentaries of the Imperial academy of Science of Petersburg.
Econophysics, 4th Year Levy stochastic process and limit theorem
Limit Theorem for Stable Distribution
As stated by control limit theorem, Gaussian distribution is an attractor
in functional space of probability density function. It is a very peculiar,
stable and infinitely divisible distribution. It is only a stable distribution
having all its moment infinite. All other stable distributions except
Gaussian have some or all their moment infinite.
There are other attractions in functional space of probability density
function’s which are non-Gaussian. There exist a limit theorem which
states that ”the probability density function of a sumof 𝑛 independent
and identically distributed random variable 𝑥
𝑖
converges on the
probability density function of the random variable 𝑥
𝑖
”. i.e., there exit
attraction basin for every form of probability density function.
Consider a stochastic process:
𝑆
𝑛
=
𝑛
𝑖=1
𝑥
𝑖
with 𝑥
𝑖
being independent and identically distributed random variable
Econophysics, 4th Year Levy stochastic process and limit theorem
Suppose
𝑝(𝑥
𝑖
)𝑁
(
𝑐
−
|𝑥
𝑖
|
−(1+𝛼)
as 𝑥 → −∞
𝑐
+
|𝑥
𝑖
|
−(1+𝛼)
as 𝑥 → +∞
and 𝛽 =
𝑐
+
− 𝑐
−
𝑐
+
− 𝑐
−
= assymetry parameter
Then
˜
𝑃(
˜
𝑆
𝑛
) = 𝑝(𝑆
𝑛
) × 𝑛
1/𝛼
approaches a stable non Gaussian
distribution p(x) of index 𝛼 and assymetry parameter 𝛽 and 𝑝(𝑆
𝑛
)
belongs to attractor basin of p(x).
𝑖.𝑒., 𝑝 (𝑆
𝑛
) → 𝑝(𝑟)
since 0 < 𝛼 ≤ 2 a infinite no. of attractor is present in the functional
shape of probability density functions. The collection of such power law
distribution comprise a family of stable distribution.
Econophysics, 4th Year Levy stochastic process and limit theorem
The main difference between Gaussian - attractor and stable
non-Gaussian attractors is that the finite variables random variables are
present in the Gaussian an basin of attraction, where as random
variables with infinite variance are characterize by distribution with
power law tails. Hence, the distribution power law tails are present in
the stable non-Gaussian basin of attraction.
Econophysics, 4th Year Levy stochastic process and limit theorem
