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From: Osher Doctorow on 28 Jul 2010 02:03
From Osher Doctorow

From the previous posts, Repulsive or Expansive sets A have the
property:

1) P(A-->A ' ) > P(A)

which is equivalent to:

2) P(A) < 1/2

while Attractive or Expansive sets A have the property:

3) P(A ' --> A) > P(A ' )

which is equivalent to:

4) P(A) > 1/2

Note that Repulsion or Expansion is equivalent to:

5) P(A --> A ' ) > P(A ' --> A)

for Repulsion/Expansion, because this is equivalent to:

6) P(A ' ) > P(A), which holds iff 1 - P(A) > P(A), which holds iff
P(A) < 1/2.

If we consider corresponding random variables X to (random) sets A,
for example A or A_x or A(x) = {w: X(w) < = x}, then there is a reason
to prefer random variables which decay slowly to infinity (among
continuous random variables), which is the case with random variables
having Memory and random variables having Fat Tails or Long Tails.
The reason is that:

7) P ' (A-->B) = 1 iff P(A) = P(B)

so that:

8) P ' (A --> A ' ) = 1 iff P(A) = P(A ' ) iff P(A) = 1 - P(A) iff
P(A) = 1/2.

Since optimality is arguably P ' (A-->B) = 1 (the maximum possible P '
which is always on a scale of 0 to 1), but the exact value of 1/2 is
not "available" due to the above equations, the closer P(A ) is to
1/2 while being < 1/2, the more optimal the Repulsion. Thus P(A) has
to be large but < 1/2, which if A is the tail of the distribution
implies that it is fat-tailed or heavy-tailed which are typically
memory distributions. But we have used the notation A to be bounded
and A ' unbounded. Since we get the same problem with A ' except for
it being > 1/2, we have to drop the boundedness conditions to make a
choice and allow A to be the tail of the distribution.

It turns out that the "contradiction" comes from the requirement that
P ' (A-->B) = 1, which also occurs from the alternative requirement
that P(A-->B) = 1. The first implies that P(A) = P(B), but since in
the definition of P ' (A-->B) we must have P(B) < = P(A), this assumes
directionality if B = A ' or if A is replaced by A ' and B by A. In
a logical sense, it requires an axiom or principle rather than a more
complicated proof.

Osher Doctorow

Osher Doctorow

From: Osher Doctorow on 28 Jul 2010 02:30
From Osher Doctorow

Before giving examples of fat-tailed/heavy-tailed random variables or
their distributions, I'll try to explain the rough intuitive meaning
to those with less probability experience. The "graph" of a
continuous random variable can very roughly often be thought of as
either a bell-shaped curve or half of a bell-shaped curve (left or
right half - let's say right half), although there are so many
distortions and exceptions that the qualifications "very roughly
often" need to be kept in mind. If the graph is roughly bell-
shaped, then the extreme left and right parts come very close to the
abscissa or x-axis (although remaining above it), and in fact approach
the x axis as x --> infinity and as x --> -infinity, and these are
called the "tails" of the random variable X or whatever symbol is used
for the random variable. If the graph is the right half of a bell-
shaped curve, then the right side gets closer and closer to the x axis
above the axis as x --> infinity, and the right side is referred to as
the "tail" of the random variable.

The tails can approach 0 in height or vertical thickness fast or
slowly. If they approach 0 slowly, then we describe the scenario as
"fat-tailed" or "heavy-tailed" or "long-tailed".

The Gaussian/normal random variables (the classical bell-shaped ones)
are not fat-tailed, but the following random variables are fat-tailed:

1) The Cauchy random variable or Cauchy distribution.
2) The Levy random variable or Levy distribution.
3) The t random variable or Student's t random variable or
distribution.

All of these are special cases of:

4) Stable distributions (stable random variables)

which readers can look up online, for example the Wikipedia articles
on them. Both the Cauchy and t random variables look bell-shaped
except their tails are fatter and they approach infinity and -infinity
slower. The Levy distribution is 1-tailed (the right tail) so it
doesn't look like a whole bell.

A complication is that the Gaussian/normal random variable is also
Stable, but it is not fat-tailed.

Look up Wikipedia's "Heavy-tailed distribution", which gives one-
tailed examples as the Pareto, log-normal, Levy, Weibull with shape
parameter < 1, Burr, Log-Gamma, and 2-tailed examples includes Cauchy,
t, and 2-tailed Stable distributions except the Gaussian/normal.

Osher Doctorow
From: Osher Doctorow on 28 Jul 2010 02:39
From Osher Doctorow

See also Wikipedia's "Power Law" which gives examples of relationships
between the above random variables (t, Cauchy, etc.) and Heavy-tailed
distributions. See also Wikipedia's "Fat tail," "Stable
distribution," etc.

Distributions or random variables with Memory are typically fat-tailed/
heavy-tailed. "Memoryless" is typically associated with the Gaussian/
normal, the Exponential, and/or Markov Chain/Markov or Markovian
(Process) scenarios. "Non-Markovian" in the literature is almost
always Memory.

Osher Doctorow
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