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Lecture 12
CLT and Levy Stable Processes
John Rundle Econophysics PHYS 250
Central Limit Theorem
https://en.wikipedia.org/wiki/Central_limit_theorem
• In probability theory, the central limit theorem (CLT) establishes that, for
the most commonly studied scenarios, when independent random
variables are added, their sum tends toward a normal distribution
(commonly known as a bell curve) even if the original variables themselves
are not normally distributed.
• In more precise terms, given certain conditions, the arithmetic mean of a
sufficiently large number of iterates of independent random variables,
each with a well-defined (finite) expected value and finite variance, will be
approximately normally distributed, regardless of the underlying
distribution.
• The theorem is a key concept in probability theory because it implies that
probabilistic and statistical methods that work for normal distributions can
be applicable to many problems involving other types of distributions.
Central Limit Theorem
https://en.wikipedia.org/wiki/Central_limit_theorem
• To illustrate the meaning of the theorem, suppose that a sample is
obtained containing a large number of observations, each observation
being randomly generated in a way that does not depend on the values of
the other observations, and that the arithmetic average of the observed
values is computed.
• If this procedure is performed many times, the central limit theorem says
that the computed values of the average will be distributed according to
the normal distribution (commonly known as a "bell curve").
• A simple example of this is that if one flips a coin many times the
probability of getting a given number of heads in a series of flips should
follow a normal curve, with mean equal to half the total number of flips in
each series.
Central Limit Theorem
https://en.wikipedia.org/wiki/Central_limit_theorem
Central Limit Theorem
http://www.slideshare.net/ShakeelNouman1/sampling-and-sampling-distributions
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
• In probability theory, a distribution or a random variable is said to be
stable if a linear combination of two independent copies of a random
sample has the same distribution, up to location and scale parameters.
• The stable distribution family is also sometimes referred to as the Lévy
alpha-stable distribution, after Paul Lévy, the first mathematician to have
studied it.
• Of the four parameters defining the family, most attention has been
focused on the stability parameter, α (see following).
• Stable distributions have 0 < α ≤ 2, with the upper bound corresponding to
the normal distribution, and α = 1 to the Cauchy distribution.
• The distributions have undefined variance for α < 2, and undefined mean
for α ≤ 1.
• The importance of stable probability distributions is that they are
"attractors" for properly normed sums of independent and identicallydistributed (iid) random variables.
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
A non-degenerate distribution is a stable distribution if it satisfies the
following property:
• Let X1 and X2 be independent copies of a random variable X. Then X is said
to be stable if for any constants a > 0 and b > 0 the random variable aX1 +
bX2 has the same distribution as cX + d for some constants c > 0 and d.
• The distribution is said to be strictly stable if this holds with d = 0.
• Since the normal distribution, the Cauchy distribution, and the Lévy
distribution all have the above property, it follows that they are special
cases of stable distributions.
• Such distributions form a four-parameter family of continuous probability
distributions parametrized by location and scale parameters μ and c,
respectively, and two shape parameters β and α, roughly corresponding to
measures of asymmetry and concentration, respectively (see the figures).
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distrib
ution
• Of the four parameters defining
the family, most attention has
been focused on the stability
parameter, α (see panel).
• Stable distributions have 0 < α ≤
2, with the upper bound
corresponding to the normal
distribution, and α = 1 to the
Cauchy distribution.
• The distributions have undefined
variance for α < 2, and undefined
mean for α ≤ 1.
• The importance of stable
probability distributions is that
they are "attractors" for properly
normed sums of independent and
identically-distributed (iid)
random variables.
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
• By the classical central limit theorem the properly normed
sum of a set of random variables, each with finite variance,
will tend towards a normal distribution as the number of
variables increases.
• Without the finite variance assumption, the limit may be a
stable distribution that is not normal.
• Mandelbrot referred to such distributions as "stable Paretian
distributions”, after Vilfredo Pareto.
• In particular, he referred to those maximally skewed in the
positive direction with 1<α<2 as "Pareto-Lévy
distributions”,which he regarded as better descriptions of
stock and commodity prices than normal distributions.
Fourier Transforms
https://en.wikipedia.org/wiki/Fourier_transform
• The Fourier transform decomposes a function of time (a
signal) into the frequencies that make it up, in a way similar to
how a musical chord can be expressed as the frequencies (or
pitches) of its constituent notes.
• The Fourier transform of a function of time itself is a complexvalued function of frequency, whose absolute value
represents the amount of that frequency present in the
original function, and whose complex argument is the phase
offset of the basic sinusoid in that frequency.
• The Fourier transform is called the frequency domain
representation of the original signal.
Fourier Transforms
https://en.wikipedia.org/wiki/Fourier_transform
• The term Fourier transform refers to both the frequency
domain representation and the mathematical operation that
associates the frequency domain representation to a function
of time.
• The Fourier transform is not limited to functions of time, but
in order to have a unified language, the domain of the original
function is commonly referred to as the time domain.
• For many functions of practical interest, one can define an
operation that reverses this, the inverse Fourier
transformation of a frequency domain representation
combines the contributions of all the different frequencies to
recover the original function of time.
Fourier Transforms
https://en.wikipedia.org/wiki/Fourier_transform
Fourier
Transforms
https://en.wikipedia.org
/wiki/Fourier_transform
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
Infinitely Divisible Processes
https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)
• In probability theory, a probability distribution is infinitely
divisible if it can be expressed as the probability distribution
of the sum of an arbitrary number of independent and
identically distributed random variables.
• The characteristic function of any infinitely divisible
distribution is then called an infinitely divisible characteristic
function.
• More rigorously, the probability distribution F is infinitely
divisible if, for every positive integer n, there exist n
independent identically distributed random variables Xn1, ...,
Xnn whose sum Sn = Xn1 + … + Xnn has the distribution F.
Infinitely Divisible Processes
https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)
• Every infinitely divisible probability distribution corresponds
in a natural way to a Lévy process. A Lévy process is a
stochastic process { Lt : t ≥ 0 } with stationary independent
increments
• Stationary means that for s < t, the probability distribution of
Lt − Ls depends only on t − s and where independent
increments means that that difference Lt − Ls is independent
of the corresponding difference on any interval not
overlapping with [s, t], and similarly for any finite number of
mutually non-overlapping intervals.
Infinitely Divisible Processes
https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)
• The Poisson distribution, the negative binomial distribution,
the Gamma distribution and the degenerate distribution are
examples of infinitely divisible distributions
• As are the normal distribution, Cauchy distribution and all
other members of the stable distribution family.
• The uniform distribution and the binomial distribution are not
infinitely divisible, as are all distribution with bounded (finite)
support.
• The Student's t-distribution is infinitely divisible, while the
distribution of the reciprocal of a random variable having a
Student's t-distribution, is not.
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
A generalized central limit theorem
• Another important property of stable distributions is the role that they
play in a generalized central limit theorem. The central limit theorem
states that the sum of a number of independent and identically
distributed (i.i.d.) random variables with finite variances will tend to a
normal distribution as the number of variables grows.
• A generalization due to Gnedenko and Kolmogorov states that the sum of
a number of random variables with symmetric distributions having powerlaw tails (Paretian tails), decreasing as |x|−α−1 where 0 < α < 2 (and
therefore having infinite variance), will tend to a stable distribution f ( x ; α
, 0 , c , 0 ) as the number of summands grows.
• If α>2 then the sum converges to a stable distribution with stability
parameter equal to 2, i.e. a Gaussian distribution.[10]
Levy Stable Processes
https://en.wikipedia.org/wiki/Stable_distribution
Levy Stable Processes - Phase Diagram
https://en.wikipedia.org/wiki/Stable_distribution
From: Martin Sewall, Characterization of Financial Time Series, UCL Research
Note RN/11/01 January 20, 2011
Comparison to Data
http://finance.martinsewell.com/stylized-facts/distribution/
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PDF of returns for the Shanghai market
data with Δt = 1 (daily returns)
This plot is compared to a stable
symmetric Levy distribution using the
value α = 1.44 determined from the slope
[in a log-log plot of the central peak of the
PDF as a function of the time increment].
Two attempts to fit a Gaussian are also
shown.
The wider Gaussian is chosen to have the
same standard deviation as the empirical
data.
However, the peak in the data is much
narrower and higher than this Gaussian,
and the tails are fatter.
The narrower Gaussian is chosen to fit the
central portion, however the standard
deviation is now too small.
It can be seen that the data has tails which
are much fatter and furthermore have a
non-Gaussian functional dependence."
Johnson, Jefferies and Hui (2003)
Comparison to Data
http://finance.martinsewell.com/stylizedfacts/distribution/